A  Manual  of 

Geometrical  Crystallography 


TREATING  SOLELY  OF  THOSE  PORTIONS 

OF   THE   SUBJECT   USEFUL   IN   THE 

IDENTIFICATION   OF   MINERALS 


BY 
G.   MONTAGUE   BUTLER,   E.M. 

\N 

Professor  of  Mineralogy  and  Petrology;  Dean,  College  of  Mines 
and  Engineering,  University  of  Arizona,  Tucson,  Arizona 


FIRST  EDITION 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:   CHAPMAN   &   HALL,    LIMITED 
1918 


COPYRIGHT,  1918 

BY 
G.  MONTAGUE  BUTLER 


Stanhope  iprcss 

F.    H.  GILSON    COMPANY 
BOSTON,  U.S.A. 


PREFACE 

CRYSTALLOGRAPHY  may  be  studied  with  two  dis- 
tinct purposes  in  view.  The  end  usually  sought  is 
the  ability  to  describe  crystals  with  such  detailed 
accuracy  that  minor  variations  between  them  and 
other  crystals  may  be  detected  and  shown  graphi- 
cally or  embodied  in  mathematical  expressions. 
This  aspect  of  the  subject  belongs  unquestionably 
in  the  realm  of  pure  science.  It  involves  the 
accurate  measurement  of  angles  with  delicate  in- 
struments requiring  careful  manipulation,  and  the 
results  secured  are  not  sought  with  any  idea  that 
they  may  have  practical  value. 

Quite  different  is  the  other  purpose  to  which 
reference  has  been  made,  since  it  is  the  attainment 
of  the  ability  to  recognize  crystal  forms  and  espe- 
cially systems  almost  instantly  with  the  use  of  few  if 
any  instruments,  and  those  who  seek  this  knowledge 
do  so  wholly  because  they  expect  to  use  it  as  a  tool 
for  identifying  minerals. 

It  is  of  the  phase  of  the  subject  last  mentioned 
that  this  book  treats,  and  it  is  hoped  that  it  will  fill 
the  needs  of  the  growing  group  of  educators  who 
realize  the  great  importance  of  teaching  "sight 
recognition"  of  minerals  to  engineering  students  or 
others  who  study  mineralogy  merely  for  its  cultural 
value.  In  order  to  conserve  the  students'  time  and 

iii 


IV  PREFACE 

energy,  everything  not  germane  to  the  end  sought  — 
the  acquisition  of  information  useful  in  the  "  sight 
recognition"  of  minerals  —  has  been  omitted,  and 
no  hesitation  has  been  felt  in  departing  from  current 
usage  when  it  seemed  desirable  in  order  to  secure 
simplicity  and  clarity. 

The  system  followed  is  not  an  untried  experiment, 
but  was  introduced  many  years  ago  by  Dr.  H.  B. 
Patton  in  the  Colorado  School  of  Mines  where  it 
has  been  taught  with  marked  success,  and  from  which 
it  has  been  carried  by  graduates  to  a  number  of 
other  institutions.  This  system  includes  the  study 
of  numerous  wooden,  cardboard,  or  plaster  models 
of  crystals,  together  with  oral  quizzes  involving  the 
instantaneous  identification  of  the  forms  repre- 
sented on  such  models,  and  discussions  of  the  theo- 
retical aspects  of  the  subject.  After  models  belong- 
ing to  a  certain  system  or  group  of  systems  have 
been  studied  and  a  sufficient  knowledge  of  them 
revealed  in  the  quiz  the  student  takes  up  the  deter- 
mination of  natural  crystals  of  the  same  degree  of 
symmetry;  and  the  study  of  crystal  models  and  of 
corresponding  natural  crystals  alternate  throughout 
the  course.  If  this  plan  is  followed,  it  will  be  neces- 
sary for  the  student  to  familiarize  himself  with  the 
matter  presented  in  Chapter  IX  before  attempting 
to  work  with  natural  crystals. 

The  author  desires  to  acknowledge  his  great 
indebtedness  to  Patton's  ''Lecture  Notes  on  Crys- 
tallography," and,  to  a  lesser  extent,  to  Bayley's 
" Elementary  Crystallography"  for  ideas  and  even 
definitions  and  descriptions  embodied  in  this  book. 
While  most  of  the  illustrations  are  original,  many  are 


PREFACE  V 

copied  without  other  acknowledgment  than  this 
from  Bayley's  "  Elementary  Crystallography/' 
Dana's  " System  of  Mineralogy,"  and  Moses  and 
Parsons'  "Mineralogy,  Crystallography,  and  Blow- 
pipe Analysis." 

G.  MONTAGUE  BUTLER 

TUCSON,  ARIZONA,  September  15,  1917. 


TABLE  OF  CONTENTS 

PAGE 

PREFACE iii 

CHAPTER  I 
INTRODUCTORY     CONCEPTIONS     AND     FUNDAMENTAL 

DEFINITIONS 1 

CHAPTER  II 

ISOMETRIC  SYSTEM 18 

Holohedral  Division 18 

Tetrahedral  Hemihedral  Division 29 

Pentagonal  Hemihedral  Division 37 

Gyroidal  Hemihedral  Division 42 

Pentagonal  Tetartohedral  Division 43 

CHAPTER  III 

HEXAGONAL  SYSTEM 45 

Holohedral  Division 45 

Rhombohedral  Hemihedral  Division 54 

Pyramidal  Hemihedral  Division 60 

Trigonal  Hemihedral  Division 64 

Trapezohedral  Hemihedral  Division 72 

Trapezohedral  Tetartohedral  Division 72 

Rhombohedral  Tetartohedral  Division 84 

CHAPTER  IV 

TETRAGONAL  SYSTEM 86 

Holohedral  Division 86 

Sphenoidal  Hemihedral  Division 93 

Pyramidal  Hemihedral  Division 97 

Trapezohedral  Hemihedral  Division . 102 

CHAPTER  V 

ORTHORHOMBIC  SYSTEM 103 

Holohedral  Division 103 

Sphenoidal  Hemihedral  Division 109 

vii 


Vlll  CONTENTS 

PAGE 
CHAPTER  VI       . 

MONOCLINIC  SYSTEM 113 

Holohedral  Division 113 

CHAPTER  VII 

TRICLINIC  SYSTEM 122 

Holohedral  Division 122 

CHAPTER  VIII 

TWINS 128 

CHAPTER  IX 

MISCELLANEOUS  FEATURES 137 

Parallelism  of  Growth 137 

Striations 138 

Cleavage 142 

Crystal  Habit 143 

Distortion 144 

Vicinal  Forms 148 

Etched  Figures  and  Corrosion 148 


vm 


CONTENTS 
CHAPTER  VI 


MONOCLINIC  SYSTEM  .  .  . 
Holohedral  Division 


CHAPTER  VII 

TRICLINIC  SYSTEM 

Holohedral  Division .  . 


PAGE 

113 
113 


122 
122 


CHAPTER  VIII 


TWINS 128 


A  Manual  of 

Geometrical  Crystallography 


CHAPTER  I 
INTRODUCTORY  CONCEPTIONS 

A  Mineral  Defined. 

(^mineral  is  a  naturally  occurring,  homogeneous, 
inorganic  substance.  / 

Diamonds,  white  mica  (muscovite),  native  gold, 
and  flint  are  illustrations  of  minerals;  while  a  silver 
coin,  granite  (consists  essentially  of  two  minerals, 
orthoclase  and  quartz,  so  is  not  homogeneous),  coal 
(is  organic  material),  and  window  glass  are  not 
minerals  although  they  belong  to  the  so-called 
" mineral  world"  as  distinguished  from  the  " animal 
and  vegetable  worlds." 

Structure  of  Minerals. 

Molecules:  Minerals,  as  well  as  all  other  sub- 
stances, are  made  up  of  extremely  small  particles 
called  molecules  which  in  any  homogeneous  sub- 
stance are  all  alike  in  composition,  size,  and  weight, 
but  which  are  unlike  in  the  particulars  mentioned  in 
different  substances.  They  are  believed  to  be  sepa- 
rated from  one  another  to  some  extent  even  in  the 
hardest  and  densest  materials.  There  is  a  tendency 

1 


CONCEPTIONS 

le  J£  be  held  m  position  with  respect 
to  adjacent  molecules  by  'certain  forces  of  mutual 
attraction,  against  which  is  opposed  a  tendency  for 
each  molecule  to  move  in  a  straight  line.  The 
relative  strength  of  these  two  tendencies  is  believed 
to  determine  whether  a  substance  is  a  gas,  a  liquid, 
or  a  solid. 

Amorphous  Structure  Defined:  (Ajsubstance  is  said 
to  possess  an  amorphous  structure  or  to  be  amor- 
phous when  its  constituent  molecules  are  arranged 
according  to  no  definite  fashion  or  pattern^  Pre- 
sumably they  lie  at  unequal  distances  with  respect 
to  each  other,  and  lines  joining  their  centers  do  not 
meet  in  fixed  angles.  If  a  box  of  oranges  be  dumped 
helter-skelter  into  a  basket  and  each  orange  be  con- 
sidered analogous  to  an  enormously  magnified  mole- 
cule, a  good  conception  of  the  structure  of  an  amor- 
phous substance  can  be  obtained. 

Natural  and  artificial  glasses  are  excellent  illus- 
trations of  amorphous  materials,  but  not  a  few 
minerals  also  possess  this  structure^ 

Crystalline  Structure  Defined:  \Jf~substance  is  said 
to  possess  a  crystalline  structure  or  to  be  crystalline 
when  the  constituent  molecules  are  arranged  in  some 
definite  fashion  or  pattern/]  A  box  of  oranges  of 
equal  size  packed  in  even  rows  and  layers  is  analo- 
gous to  a  crystalline  substance  in  which  each  orange 
corresponds  to  a  molecule,  but  it  should  not  be 
understood  that  all  crystalline  materials  have  a 
structure  resembling  in  detail  the  illustration  just 
given. 

Crystalline  substances,  while  they  may  resemble 
amorphous  ones  very  closely,  at  least  superficially, 


INTRODUCTORY  CONCEPTIONS  3 

can  usuallybe  recognized  by  the  presence  of  cleavage 
"(see"p.  l42yoTdistinctive  optical,  elecTncalTtniermaX 
or  other  physical  properties  which  prove  that  in 
crystalline  materials  there  are  certain  directions 
along  which  forces  or  agents  act  with  quite  different 
effects  from  those  produced  in  other  directions. 
\Thus  a  sphere  of  glass  (amorphous)  when  heated 
expands  equally  in  all  directions  and  remains  per- 
fectly spherical;  while  a  sphere  of  emerald  (crys- 
talline) if  similarly  heated  will  be  distortecjand  will 
become  ellipsoidal  due  to  the  fact  that  the  coefficient 
of  expansion  in  one  direction  differs  from  that  in  all 
others. 

The  majority  of  minerals  as  well  as  many  artificial 
substances  have  crystalline  structures. 

A  most  useful  characteristic  of  a  crystalline  sub- 
stance results  from  the  fact  that  at  the  time  of  its 
formation  it  shows  a  more  or  less  pronounced 
tendency  to  form  a  body  bounded  wholly  or  partially 
by  plane  surfaces  or  faces.  Such  a  partial  or  com- 
plete polyhedron  is  called  a  crystal;  and,  if  several 
such  crystals  develop  in  contact  with  or  close  proxim- 
ity to  one  another,  a  group  of  crystals  results. 

While  no  simple  definition  distinguishing  between 
single  crystals  and  crystal  groups  can  be  offered, 
there  should  be  little  chance  of  a  misconception 
arising  through  the  use  of  the  following  definition. 

A  Crystal  Defined. 

(X  crystal  is  a  crystalline  substance  bounded 
wholly  or  partially  by  natural  plane  surfaces  called 
faces  which  have  not  been  produced  by  external 
forcesj 


4  INTRODUCTORY  CONCEPTIONS 

From  what  has  been  said  it  must  be  evident  that 
a  crystal  always  has  a  crystalline  structure,  but  it  is 
equally  important  to  remember  that  crystalline  sub- 
stances do  not  by  any  means  always  occur  in  crys- 
tals. These  are  the  exception  rather  than  the  rule, 
and  develop  only  when  conditions  are  favorable. 
When  faces  are  lacking,  other  features  (such  as  the 
presence  of  cleavage)  or  physical  tests  must  be  used 
to  determine  whether  a  substance  is  crystalline  or 
amorphous.  _^ 

Formation  of  Crystals:  >Ckystals  may  form  in  two 
ways,  namely,  through  deposition  from  solutions 
(including  fusions  which  are  now  recognized  as  forms 
of  solution)  and  from  the  sublimed  (gaseous)  condi- 
tion^? In  either  case,  a  solid  molecule  having  formed, 
growth  occurs  through  the  addition  of  myriads  of 
other  molecules  which  surround  the  first  according 
to  some  definite  geometric  plan.  If  the  resulting 
crystal  is  in  suspension  in  a  gas,  vapor,  or  liquid,  it 
will  be  entirely  bounded  by  crystal  faces;  otherwise, 
only  those  portions  that  are  surrounded  by  the  gas, 
vapor,  or  liquid  will  develop  in  the  manner  outlined. 

Crystals  are  often  formed  in  the  manufacture  of 
artificial  substances,  and  these  are  subject  to  the 
same  laws  that  apply  to  mineral  crystals. 

Crystallography  Defined. 

^Crystallography  is  the  science  that  deals  with 

crystals.  J 

Three  branches  of  this  science  are  recognized. 
These  are  geometrical  crystallography,  physical 
crystallography,  and  chemical  crystallography.  The 
scope  of  each  is  suggested  by  its  name.  The  student 


FUNDAMENTAL  DEFINITIONS  5 

of  determinative  mineralogy  is  most  concerned  with 
the  first  of  these  branches,  and  this  manual  deals 
almost  entirely  with  that  phase  of  the  science. 

The  study  of  crystals  has  great  practical  value  to 
a  mineralogist  since  it  has  been  found  that  each 
crystalline  mineral  occurs  in  crystals  whose  shapes 
resemble  each  other  very  closely,  and  are,  indeed, 
frequently  almost  identical  no  matter  where  found. 
Further,  it  is  true  that  crystals  of  different  minerals 
are  usually  quite  dissimilar,  and  it  is  often  possible 
for  one  familiar  with  crystallography  to  distinguish 
easily  between  two  crystallized  minerals  which, 
except  for  the  difference  in  their  crystals,  resemble 
each  other  very  closely.  Crystallographic  terms  are 
also  employed  in  describing  features  used  as  criteria 
in  determinative  mineralogy? 

FUNDAMENTAL    DEFINITIONS 

Some  of  the  definitions  that  follow  apply  only  to 
geometrically  perfect  crystals  or  crystal  models.  In 
the  cases  of  incomplete  and  distorted  crystals  (dis- 
cussed later)  these  conceptions  will  have  to  be 
modified  as  suggested  in  the  concluding  chapter. 

A  Symmetry  Plane  Defined. 

A  symmetry  plane  is  any  plane  which  divides  an 
object  in  such  a  way  that  any  line  drawn  perpen- 
dicular thereto,  if  extended  in  both  directions,  will 
strike  the  exterior  of  the  object  in  similar  points 
which  are  equidistant  from  Ijie  dividing  plane. 

Thus,  in  Fig.  1,  A  A'  is  a  symmetry  plane  because 
a  perpendicular  drawn  to  it  at  any  point,  as  at  B, 


6  INTRODUCTORY  CONCEPTIONS 

strikes  the  exterior  at  C  and  at  C"  which  are  similar 
points  equidistant  from  the  plane.  MM'  is  not  a 
symmetry  plane,  however,  since  a  perpendicular 
erected  to  it  at  N  strikes  the  exterior  at  0  and  0' 

. 
I        A 


\  \ 

/ 

r/ 

X' 

\ 

\ 

b'      ; 

FIG 

\!       Z' 
.  1. 

which  are  neither  similar  points  (one  is  at  a  corner 
and  the  other  lies  on  an  edge)  nor  are  they  equi- 
distant from  the  plane  under  consideration.  It 
should  be  observed,  however,  that  the  perpendicular 
erected  at  B  strikes  the  exterior  at  Z  and  Z'  which 
are  similar  points  and  are  equidistant  from  MM'. 
The  definition  states,  however,  that,  in  order  that 
a  given  plane  shall  be  a  symmetry  plane,  the  test 
given  must  be  applicable  to  any  perpendicular  one 
chooses  to  select,  and  it  has  already  been  shown  that 
it  does  not  hold  in  the  case  of  00'. 

The  object  used  in  the  illustrations  just  given  is  a 
surface,  but  the  same  considerations  apply  to  sym- 
metry planes  in  solids. 

^^Lnother  definition  of  a  symmetry  plane  especially 
useful  in  the  case  of  solids  is  the  following: 

A  symmetry  plane  is  any  plane  so  situated  that, 
if  it  were  a  mirror,  the  reflection  of  the  portion  in 


FUNDAMENTAL  DEFINITIONS  7 

front  of  the  mirror  would  seem  to  coincide  exactly 
with  the  part  behincT 

From  what  has  beerTsaid  it  is  easy  to  see  that, 
while  symmetry  planes  divide  objects  into  halves 
identical  in  shape  and  size,  the  mere  fact  that  an 
object  is  so  divided  does  not  prove  that  the  dividing 
plane  is  a  symmetry  plane.  In  Fig.  1  plane  MM' 
divides  the  object  into  equal  halves,  but  is  not  a 
symmetry  plane. 

As  a  corollary  of  the  foregoing,  it  may  be  said 
that  a  symmetry  plane  is  any  plane  that  divides  an 
object  in  such  a  way  that  every  edge,  corner,  and 
face  on  one  side  of  the  plane  is  exactly  balanced  by 
identical  edges,  corners,  and  faces  directly  opposite 
on  the  other  side  of  the  plane.  * 

All  symmetry  planes  may  be  called  either  principal 
symmetry  planes  or  secondary  (sometimes  called 
common)  symmetry  planes,  as  is  explained  later. 

A  Symmetry  Axis  Defined. 

"A  symmetry  axis  is  the  line  or  direction  perpen- 
dicular to  a  symmetry  plane  and  passing  through 
the  center  of  the  object. 

A  Principal  Symmetry  Plane  Denned. 

A  principal  symmetry  plane  is  a  symmetry  plane 
perpendicular  to  which  lie  at  least  two  interchangeable 
symmetry  planes  (either  principal  or  secondary). 

It  should  be  remembered  that  there  are  three 
parts  to  this  definition,  and  that  any  principal 
symmetry  plane  must  conform  to  all  of  them. 
First,  it  must  divide  an  object  symmetrically  —  be 
a  symmetry  plane  as  already  defined.  Second,  at 


8  INTRODUCTORY  CONCEPTIONS 

least  two  other  symmetry  planes  existing  in  the 
object  must  be  perpendicular  to  the  plane  under 
consideration  (these  need  not  be  perpendicular  to 
each  other).  Third,  the  two  or  more  symmetry 
planes  perpendicular  to  the  one  under  consideration 
must  be  interchangeable.  This  third  condition  is 
the  one  most  frequently  misunderstood  or  over- 
looked by  beginners.  Attention  should,  then,  be 
especially  directed  to  the  following  paragraph. 

Interchangeable  Symmetry  Planes  and  Axes  Defined. 
Two  symmetry  planes  or  axes  are  said  to  be  inter- 
changeable when  one  plane  or  axis  may  be  placed  in 
the  position  of  the  other  plane  or  axis  without 
apparently  altering  the  appearance  or  position  of 
the  object. 

A  Principal  Symmetry  Axis  Defined. 

A  principal  symmetry  axis  is  a  symmetry  axis 
perpendicular  to  a  principal  symmetry  plane. 

A  Secondary  (or  Common)  Symmetry  Plane  Defined. 
A  secondary  symmetry  plane  is  any  symmetry 
plane  that  does  not  possess  the  characteristics  of  a 
principal  symmetry  plane  as  already  defined. 

A  Secondary  Symmetry  Axis  Defined. 

A  secondary  symmetry  axis  is  a  symmetry  axis 
perpendicular  to  a  secondary  symmetry  plane. 

An  Interfacial  Angle  Defined. 

An  interfacial  angle  is  an  angle  formed  at  the 
intersection  of  two  faces  or  the  planes  of  two  faces. 
It  must  be  measured  perpendicular  to  the  edge 


FUNDAMENTAL  DEFINITIONS  9 

formed  by  the  intersection  of  the  two  faces;  or,  if 
the  faces  do  not  intersect  in  an  edge,  the  measure- 
ment must  be  made  perpendicular  to  the  imaginary 
line  located  at  the  intersection  of  the  planes  of  the 
two  faces. 

A  Zone  Defined. 

A  zone  is  a  group  of  faces  in  the  form  of  a  belt  or 
band  which  extends  around  a  crystal  in  such  a  way 
that  the  edges  formed  by  the  mutual  intersections 
of  the  faces  are  all  parallel. 

A  Zonal  Axis  Denned. 

A  zonal  axis  is  a  line  through  the  center  of  a 
crystal  parallel  to  the  faces  of  a  zone. 

Replaced  Edges  and  Corners  Denned. 

A  face  is  said  to  replace  an  edge  when  that  face  is 
substituted  for,  and  lies  parallel  to,  the  edge,  yet  is 
not  equally  inclined  to  the  two  faces  whose  inter- 
section would  form  that  edge. 

Similarly,  a  face  may  be  said  to  replace  a  corner 
formed  by  the  intersection  of  three  or  more  faces 
when  it  is  substituted  for  that  corner,  but  is  not 
equally  inclined  to  at  least  one  set  of  similar  faces 
whose  intersection  would  form  that  corner. 

Truncated  Edges  and  Corners  Denned. 

A  face  is  said  to  truncate  an  edge  when  it  is  sub- 
stituted for  that  edge  in  such  a  way  as  to  be  parallel 
to  it  and  to  make  equal  angles  with  the  faces  whose 
itersection  would  form  that  edge.     Fig.  2  shows  a 
with  truncated  edges. 


10 


INTRODUCTORY  CONCEPTIONS 


Similarly,  a  face  may  be  said  to  truncate  a  corner 
when  it  is  substituted  for  that  corner  and  makes 
equal  angles  with  all  similar  faces 
whose  intersections  would  form 
that  corner.  Fig.  2  shows  a  crys- 
tal with  truncated  corners. 


FIG.  2.  —  Hexahe- 
dron (cube)  with 
edges  truncated  by 
the  dodecahedron, 
and  corners  truncated 
by  the  octahedron. 


Beveled  Edges  Defined. 

Two  faces  are  said  to  bevel  an 
edge  if  they  replace  the  edge  in 
such  a  way  that  equal  angles  are 
formed  between  each  replacing 
face  and  the  adjacent  faces 
whose  intersection  would  form 
the  edge.  Fig.  3  shows  a  crystal 
with  beveled  edges. 


A  Crystal  System  Defined. 

All  those  crystals  which  con-  FIG.  3.  —  Hexahe- 
tain  the  same  number  and  kind  dron  (cube)  with 

of    symmetry    planes    (together  ed&es  beve11^  by  the 

.,,  ,         ,     ,         ,,      tetrahexahedron. 

with    others    produced    by    the 

suppression  of  certain  faces  in  accordance  with 
definite  laws,  which  may  be  regarded  as  modifi- 
cations of  these)  are  said  to  belong  to  the  same 
crystal  system. 

Number  and  Names  of  the  Crystal  Systems. 

Any  crystal  may  be  placed  in  one  of  six  crystal 
systems.  Of  these  there  are  32  subdivisions  or 
classes,  but  few  or  no  minerals  are  known  to  occur 
in  some  of  these,  so  familiarity  with  all  of  them  is 
unnecessary. 


FUNDAMENTAL  DEFINITIONS 
The  six  crystal  systems  are  as  follows : 1 


11 


Number  and  kind  of  symmetry  planes  in 
crystals  with  the  fully  developed 
symmetry  of  the  system. 

Name  of  the  system. 

Principal. 

Secondary. 

3 

6 

Isometric 

1 
1 

6 
4 

Hexagonal 
Tetragonal 

0 
0 
0 

3 
1 

0 

Orthorhombic 
Monoclinic 
Triclinic 

Any  one  crystalline  mineral  species  always  occurs 
in  crystals  characterized  by  the  presence  of  a  definite 
number  and  kind  of  symmetry  planes,  which  is 
equivalent  to  saying  that  its  crystals  always  belong 
to  a  certain  one  of  the  subdivisions  of  the  above- 
mentioned  crystal  systems.  Crystals  containing  the 
same  kind  and  number  of  symmetry  planes  are  said 
to  show  the  same  degree  of  symmetry. 

1  As  it  is  absolutely  necessary  that  the  student  should  be 
able  to  recognize  symmetry  planes  and  to  distinguish  be- 
tween principal  and  secondary  symmetry  planes,  it  is  desir- 
able that,  before  proceeding  further,  a  number  of  crystal 
models  should  be  separated,  first  into  three  groups  each  con- 
taining the  same  number  of  principal  symmetry  planes,  and 
then  into  the  six  crystal  systems,  to  do  which  the  number 
of  secondary  symmetry  planes  must  also  be  considered. 
Wooden  models  for  this  work  are  manufactured  by  Dr.  F. 
Krantzr,  Bonn  on  Rhine,  Germany,  and  sold  at  an  average 
price  of  less  than  $0.50  each. 


12  INTRODUCTORY  CONCEPTIONS 

Designating  the  Position  of  Planes  in  Space. 

The  position  of  any  plane  may  be  defined  by 
ascertaining  its  relation  to  three  fixed  lines  or  axes 
intersecting  in  a  common  point  called  the  origin. 
This  may  be  done  by  determining  the  distance  and 
direction  from  the  origin  to  the  point  at  which  the 
plane  cuts  each  axis.  As  crystals  are  bounded  by 
faces  which  are  circumscribed  portions  of  planes, 
the  positions  of  such  faces  may  be  given  by  referring 
them  to  such  a  system  of  axes. 

Crystal  Axes  Defined. 

Crystal  axes  are  fixed  lines  or  directions  to  which 
crystal  faces  are  referred  for  the  purpose  of  ascer- 
taining their  mutual  relationships. 

\J  General  Rule  for  Choosing  Crystal  Axes. 

Select  the  crystal  axes  so  that  they  coincide  where 
possible  with  symmetry  axes,  giving  the  preference 
to  principal  symmetry  axes;  but  when  an  insuffi- 
cient number  of  symmetry  axes  are  present  choose 
lines  passing  through  the  center  of  the  crystal  and 
parallel  to  prominent  crystallographic  directions, 
preferably  edges.  The  crystal  axes  should  intersect 
at  as  nearly  right  angles  as  possible  in  all  systems 
but  the  hexagonal  in  which  it  is  convenient  to  have 
the  horizontal  axes  intersect  at  angles  of  60°  and  120°, 
instead  of  90°. 

The  Terms  Crystal  and  Symmetry  Axes  not  Synony- 
mous. 

Care  should  be  exercised  not  to  confuse  the  terms 
crystal  axes  and  symmetry  axes.  While  it  is  true 


FUNDAMENTAL  DEFINITIONS  13 

that  these  sometimes  coincide,  they  do  not  by  any 
means  always  do  so.  Some  symmetry  axes  are 
never  used  as  crystal  axes;  and  many  crystal  axes 
are  not  symmetry  axes  at  all. 

Designation  and  Use  of  Crystal  Axes. 

It  is  customary  to  call  the  crystal  axis  extending 
from  front  to  back  the  a  axis,  the  one  from  right  to 
left  the  6  axis,  and  the  vertical  one  the  c  axis.  Inter- 
changeable crystal  axes  are  represented  by  the  same 
letter,  however. 

It  will  later  be  made  plain  that  the  faces  on  any 
crystal  may  be  separated  into  groups  each  of  which 
is  characterized  by  the  fact  that  its  faces  (or  the 
planes  of  the  faces)  intersect  all  the  crystal  axes  at 
distances  from  the  origin  which  bear  the  same  fixed 
ratios  to  each  other.  It  has  been  observed  that,  in 
the  case  of  any  given  mineral  species,  certain  such 
groups  are  comparatively  common  while  others  are 
less  common  or  rare.  It  is  the  usual  presence  of 
certain  such  groups  of  faces  that  causes  the  crystals 
of  any  given  species  to  resemble  each  other  very 
closely,  and  makes  it  possible  to  classify  instantly 
many  minerals  occurring  in  crystals. 

Ground-form  or  Unit-form  Denned. 

The  ground-form  or  unit-form  of  the  crystals  of 
any  mineral  species  is  the  most  commonly  occurring 
group  of  faces  that  intersect  all  the  crystal  axes  at 
finite  distances  from  the  origin  which  distances  bear 
the  same  fixed  ratios  to  each  other.  In  the  iso- 
metric system  the  octahedron  (see  p.  19)  is  called 
the  ground-form. 


14  INTRODUCTORY  CONCEPTIONS 

Unit  Axial  Lengths  Denned. 

The  distances  from  the  origin  at  which  the  faces 
(or  faces  extended)  of  the  ground-form  intersect  the 
crystal  axes  are  considered  the  unit  axial  lengths 
of  those  axes,  provided  that  such  a  scale  be  used 
as  will  make  the  length  of  at  least  one  of  these 
axial  lengths  unity.  This  definition  applies  to  all 
systems  but  the  isometric.  In  that  system  the 
unit  axial  length  is  the  shortest  one  of  the  three 
distances  measured  from  the  origin  to  the  points 
where  a  face  (or  the  plane  of  a  face)  intersects  the 
crystal  axes. 

Practically,  of  course,  any  scale  can  be  used  in 
making  the  measurements  mentioned  in  the  last 
paragraph,  since,  for  instance,  if  the  distances 
measured  in  any  scale  on  the  a,  6,  and  c  axes  are, 
respectively,  1.817,  1.112,  and  1.253,  and  it  is 
desired  to  have  the  b  axial  length  unity,  this  may  be 
brought  about  without  affecting  the  ratio  between 
the  expressions  by  dividing  each  of  the  three  expres- 
sions by  the  value  for  b.  If  this  is  done,  the  results 
will  be  1.634,  1.000,  and  1.125  which  are  the  unit 
axial  lengths  of  the  crystals  of  the  mineral  selected 
as  an  illustration  (sylvanite). 

The  letters  a,  b,  and  c  are  used  not  only  to  desig- 
nate the  crystal  axes,  but  also  to  represent  the  unit 
axial  lengths  of  these  axes. 

The  distances  from  the  origin  at  which  faces  (or 
faces  extended)  other  than  those  belonging  to  the 
ground-form  cut  the  axes  are  expressed  in  terms  of 
the  unit  axial  lengths,  as  4a,  26,  and  Ic.  If  it  is 
desired  to  have  unity  for  the  coefficient  of  b,  this 
may  be  secured  by  dividing  each  expression  by  the 


FUNDAMENTAL  DEFINITIONS  15 

coefficient  of  6,  which  reduces  the  expressions  to  2a, 
16,  and  f  c. 

A  Parameter  Defined. 

A  parameter  is  the  distance  from  the  origin  to  the 
point  where  a  face  (or  a  face  extended)  cuts  a  crystal 
axis,  measured  in  terms  of  the  unit  length  of  that 
axis.  Thus,  in  the  illustration  given  in  the  preced- 
ing paragraph,  2,  1,  and  |  are  the  parameters  of  the 
face  under  consideration  on  the  a,  6,  and  c  axis, 
respectively.  It  is  customary  to  use  m,  n,  and  p  as 
general  expressions  for  parameters.  A  face  parallel 
to  an  axis  will  intersect  that  axis  at  infinity,  and  will 
have  infinity  (oo )  for  its  parameter  on  that  axis. 

The  Law  of  Rationality  of  Parameters. 

Parameters  are  always  rational,  fractional  or 
whole,  small  or  infinite  numbers. 

Crystal  Form  Defined. 

A  crystal  form  is  a  group  of  faces  with  identical 
parameters  all  of  which  are  necessary  to  complete 
the  symmetry  of  the  system. 

In  explanation  of  this  definition,  it  may  be  stated 
that  in  studying  any  system  of  crystals,  if  we  assume 
the  presence  of  a  face  or  plane  of  given  parameters, 
there  must  be  present  a  definite  number  of  other 
faces  with  identically  the  same  parameters  in  order 
that  the  complete  symmetry  of  the  system  may  be 
shown.  Such  a  group  of  faces  is  technically  known 
as  a  crystal  form.  It  will  later  be  shown  that  there 
are  seven  distinctly  different  forms  in  each  system 
and  in  each  subdivision  of  a  system. 


16  INTRODUCTORY  CONCEPTIONS 

Crystal  Form  and  Shape  Differentiated. 

The  student  should  be  careful  not  to  confuse  the 
terms  "form"  and  "shape"  as  applied  to  crystals. 
A  crystal  may  have  the  general  appearance  of  a 
cube,  for  instance,  yet  bear  no  faces  with  the  param- 
eters characteristic  of  the  crystal  form  known  as 
the  cube.  It  may  still,  with  propriety,  be  said  to 
have  a  cubical  shape,  although  the  crystal  form 
known  as  the  cube  is  not  represented  upon  it, 

Symbol  Defined. 

A  symbol  in  the  Weiss  system,  of  which  a  slight 
modification  is  used  in  this  book,  is  the  product  of 
the  parameters  of  a  face  and  the  corresponding  unit 
axial  lengths,  arranged  in  the  form  of  a  ratio,  as 
na  :  b  :  me. 

Since  every  face  of  any  one  form  has  the  same 
parameters  and  unit  axial  lengths,  it  follows  that  the 
symbols  of  any  face  may  be  regarded  as  the  symbol 
of  the  form  of  which  that  face  is  a  part. 

Several  other  systems  of  symbols  are  in  more  or 
less  widespread  use,  and  are  presented  in  the  more 
extended  textbooks  on  crystallography.  Lists  of 
such  symbols  without  further  explanation  are 
in  this  book  appended  to  the  description  of  each 
crystal  system. 

Law  of  Axes. 

The  opposite  ends  of  crystal  axes  (as  well  as  of 
symmetry  axes)  must  be  cut  by  the  same  number  of 
similar  crystal  faces  similarly  arranged. 

The  importance  of  this  law  will  be  understood  when 
the  monoclinic  and  triclinic  systems  are  studied. 


FUNDAMENTAL  DEFINITIONS  17 

Holohedral,    Hemihedral,    and   Tetartohedral    Forms 
Defined. 

Holohedral  Forms:  When  a  form  has  the  full 
symmetry  of  the  system  to  which  it  belongs  (see 
p.  11)  it  is  said  to  be  holohedral. 

Hemihedral  (half)  Forms:  A  hemihedral  form  may 
be  conceived  to  be  developed  by  dividing  a  holo- 
hedral form  by  means  of  a  certain  set  or  sets  of 
symmetry  planes  into  a  number  of  parts,  then 
suppressing  all  feces  lying  wholly  within  alternate 
parts  thus  obtained,  and  extending  all  the  remaining 
faces  until  they  meet  in  edges  or  corners. 

Tetartohedral  (quarter)  Forms:  Tetartohedral  forms 
may  be  conceived  to  be  developed  from  holohedral 
ones  by  the  simultaneous  application  of  two  different 
types  of  hemihedrism.  These  may  be  regarded  as 
the  half  forms  of  half  forms. 

Hemimorphic  Crystals  Denned. 

A  hemimorphic  crystal  is  one  in  which  the  law  of 
axes  is  violated  so  far  as  one  crystal  axis  is  con- 
cerned; that  is,  the  opposite  ends  of  one  crystal 
axis  are  not  cut  by  the  same  number  of  similar  faces 
similarly  arranged. 

Comparatively  few  minerals  occur  as  hemimorphic 
crystals. 


CHAPTER  II 
ISOMETRIC  SYSTEM 

HOLOHEDRAL  DIVISION 

Symmetry. 

The  holohedral  division  of  the  isometric  system  is 
characterized  by  the  presence  of  three  interchange- 
able principal  symmetry  planes  and  six  interchange- 
able secondary  symmetry  planes.  The  former 
intersect  at  angles  of  90°,  and  the  latter  at  60°,  90°, 
and  120°  angles.  The  two  classes  of  symmetry 
planes  are  so  arranged  that  every  90°  angle  between 
principal  symmetry  planes  is  bisected  by  a  secondary 
symmetry  plane. 

The  Selection,  Position,  and  Designation  of  the  Crystal 
Axes. 

In  accordance  with  the  general  rule  (see  p.  12), 
the  crystal  axes  in  the  isometric  system  are  chosen 
so  as  to  coincide  with  the  principal  symmetry  axes. 
There  are,  then,  in  the  isometric  system  three  inter- 
changeable crystal  axes  which  are  at  right  angles  to 
each  other.  One  is  held  vertically,  and  one  so  as  to 
extend  horizontally  from  right  to  left;  the  other 
must  then  extend  horizontally  from  front  to  back. 
Since  all  of  these  axes  are  mutually  interchangeable, 

each  is  called  an  a  axis. 

i  ~&~\ 

I      When  a  crystal  is  so  held  that  the  crystal  axes 
/  extend  in  the  proper  direction  as  viewed  by  the! 
\  observer  it  is  said  to  be  oriented. 
V—  18 


HOLOHEDRAL  DIVISION 


19 


Orienting  Crystals. 

Holohedral  isometric  crystals  are  oriented  by 
holding  a  principal  symmetry  plane  so  that  it  ex- 
tends vertically  from  front  to  back;  then  rotating 
the  crystal  around  the  principal  symmetry  axis 
perpendicular  to  this  plane  until  another  principal 
symmetry  plane  extends  vertically  from  right  to 
left,  and  a  third  such  plane  lies  horizontally.  The 
crystal  axes  will  then  extend  in  the  proper  directions. 

An  Octant  Denned. 

An  octant  in  all  systems  but  the  hexagonal  is  one 
of  the  eight  parts  obtained  by  dividing  a  crystal  by 
means  of  three  planes  each  of  which  contains  two 
crystal  axes. 

Holohedral  Isometric  Forms  Tabulated.  ]- 

The  holohedral  isometric  forms,  together  with  other 
data  relating  to  each,  are  given  in  the  following  table : 


Symbol. 

Name. 

Num- 
ber of 

faces. 

Three  axes  cut  alike 

a    a:  a 

Octahedron  (Fig.  4) 

8 

Two  axes 
cut  alike 

Two  axes  cut  at 
a  distance  < 
the  other 

\     \ 
a    a  :  ma 
a    a  :  ooa 

Trisoctahedfon  (Fig.  5) 
Dodecahedron  (Fig.  6) 

24 
12 

Two  axes  cut  at 
a  distance  > 
the  other 

a    ma  :  ma 

a    oo  a  :  oo  a 
1     o       0 

Trapezohedron  (Fig.  7) 
Hexahedron  (cube) 
(Fig.  8) 

24 
6 

Three  axes  cut  unlike 

a    ma  :  na 
a    ma  :  oo  a 

Hexoctahedron  (Fig.  9) 
Tetrahexahedron  (Fig.  10) 

48 
24 

Notes.  —  In  the  isometric  system,  m  and  n  are  never  less  than  unity. 

In  the  isometric  system,  it  is  customary  to  abbreviate  the  sym- 
bols by  omitting  the  letter  a  and  the  ratio  sign  as  Imn  for 
a  '.  ma  :  na. 

The  symbol  a  :  a  :  ma,  for  instance,  is  read  a,  a,  ma  without  men- 
tion of  the  proportion  signs. 


20 


ISOMETRIC  SYSTEM 


FIG.  4.  —  Octahedron.  FIG.  5.  —  Trisoctahedron. 


FIG.  6.  —  Dodecahedron.  FIG.  7. —  Trapezohedron. 


FIG.  8.  —  Hexahedron  (cube).      FIG.  9.  —  Hexoctahedron. 


FIG.  10.  —  Tetrahexahedron. 

Synonyms  for  the  Names  of  the  Holohedral  Isometric 
Forms. 

Octahedron  —  none. 

Trisoctahedron  —  trigonal  trisoctahedron. 
Dodecahedron  —  rhombic  dodecahedron. 
Trapezohedron  —  icositetrahedron   or   tetragonal 
trisoctahedron. 


HOLOHEDRAL  DIVISION  21 

Hexahedron  —  cube. 
Hexoctahedron  —  none. 
Tetrahexahedron  —  none. 

Method  of  Determining  Holohedral  Isometric  Forms 
by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  select  any  face  in  the  upper  right 
octant  facing  the  observer,  and  ascertain  the  relative 
distances  from  the  origin  at  which  its  plane  cuts  the 
three  crystal  axes.  This  may  be  done  mentally  or 
by  laying  a  card  upon  the  face  and  using  a  pencil  to 
represent  each  axis  in  turn.  If  it  appears  that  the 
three  axes  are  cut  equally,  the  symbol  of  the  face 
(and  of  the  form  of  which  it  is  a  part)  is  a  :  a  :  a. 
By  referring  to  the  table  given  on  p.  19,  which 
should  be  memorized  as  soon  as  possible,  it  is  seen 
that  the  form  is  the  octahedron.  Similarly,  if 
the  plane  of  the  face  cuts  one  axis  comparatively 
near  the  center  of  the  crystal,  and  the  other  two 
at  greater,  but  equal,  distances,  the  symbol  is 
a  :  ma  :  ma,  and  the  form  represented  is  the  trapezo- 
hedron. 

If  more  than  one  form  is  represented  on  the 
crystal  (see  p.  25),  each  may  be  determined  in  the 
same  way.  By  using  this  method,  it  may  be  found 
that  a  crystal  like  Fig.  2  shows  the  hexahedron, 
dodecahedron,  and  octahedron;  and,  no  matter  how 
complex  a  crystal  may  be,  the  forms  represented 
upon  it  may  thus  be  readily  ascertained. 

After  the  name  of  a  form  has  been  determined  by 
the  method  suggested  the  result  may  be  checked  by 
noting  whether  the  form  in  question  has  the  requisite 


22  ISOMETRIC  SYSTEM 

number  of  faces.  To  determine  the  number  of 
faces,  it  is  only  necessary  to  count  those  lying  in  one 
octant  and  to  multiply  this  sum  by  eight.  If  it  is 
found,  for  instance,  that  three  half  (1J)  faces  lie 
within  an  octant,  the  form  has  eight  times  one-and- 
a-half,  or  twelve  faces,  and  is  a  dodecahedron. 

•''.     Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

While  it  is  permissible  and,  in  fact,  almost  neces- 
sary at  first  to  use  the  symbols  for  the  purpose  of 
determining  crystal  forms,  this  method  is  too  slow 
to  be  wholly  satisfactory,  and  should  quickly  be 
displaced  by  the  one  outlined  below,  which  has  for 
its  aim  the  instantaneous  determination  of  forms 
through  familiarity  with  the  position  or  slope  of  one 
or  more  of  their  faces. 

In  order  to  use  this  method,  orient  the  crystal, 
and  study  the  face  or  faces  of  different  shape  or  size 
in  or  near  the  upper  right  octant  facing  the  observer. 
Then  determine  which  of  the  following  descriptions 
(which  should  be  learned  at  once)  apply  to  the  face 
or  faces  seen. 

Cube:  A  horizontal  face  on  top  of  the  crystal. 
The  faces  of  the  cube  are  parallel  to  the  principal 
symmetry  planes. 

Dodecahedron:  A  face  parallel  to  the  right  and  left 
axis  and  sloping  down  toward  the  observer  at  a  steep 
angle  —  45°  from  the  horizontal.  The  dodecahedron 
has  three  faces  lying  in  the  octant  with  an  edge 
running  from  above  the  center  of  the  octant  up  to- 
ward the  vertical  axis,  but  these  faces  do  not  lie 
wholly  within  the  octant.  (Compare  with  the  tris- 


HOLOHEDRAL  DIVISION  23 

octahedron.)     The  faces  of  the  dodecahedron  are 
parallel  to  the  secondary  symmetry  planes. 

Tetrahexahedron:  A  face  parallel  to  the  right  and 
left  axis  and  sloping  down  toward  the  observer  at 
a  relatively  gentle  angle  —  less  than  45°  from  the 
horizontal.  The  tetrahexahedron  has  six  faces  lying 
in  the  octant,  but  they  do  not  lie  wholly  within  the 
octant.  (Compare  with  the  hexoctahedron.) 

Octahedron:    A  single  face  in  the  center  of  the 
octant,  sloping  steeply  down  from  the  vertical  axis  — 
at  an  angle  of  54f  °  with  the  horizontal. 

Trapezohedron:  Three  faces  lying  wholly  within 
the  octant  and  so  arranged  that  a  face  slopes  above 
the  center  of  the  octant  up  toward  the  vertical  axis 
at  an  angle  less  steep  than  that  shown  by  the  octa- 
hedron. It  is  often  useful  to  remember,  further, 
that  two  faces  forming  part  of  the  same  trapezo- 
hedron  may  intersect  below  the  center  of  the  octant  in 
an  edge  that  points  directly  toward  the  vertical  axis. 

The  trapezohedron  is  most  apt  to  be  confused 
with  the  trisoctahedron,  described  below,  and  the 
two  descriptions  should  be  carefully  compared. 

Trisoctahedron:  Three  faces  lying  wholly  within 
the  octant  and  so  arranged  that  in  the  unmodified 
form  an  edge  slopes  above  the  center  of  the  octant  up 
toward  the  vertical  axis.  Even  when  so  modified 
that  the  edge  is  lacking,  it  is  easy  to  see  that  two 
faces  extended  would  intersect  in  such  an  edge. 

Hexoctahedron:  Six  faces  lying  wholly  within  the 
octant.  As  with  the  trisoctahedron  and  dodeca- 
hedron, there  is,  on  the  unmodified  form,  an  edge 
running  above  the  center  of  the  octant  up  toward 
the  vertical  axis. 


24  ISOMETRIC  SYSTEM 

Fixed  and  Variable  Forms  Defined. 

A  Fixed  Form:  A  fixed  form  is  one  that  has  no 
variable  parameter  (m,  n,  or  p)  in  its  general  symbol. 
The  octahedron,  dodecahedron,  and  cube  are  fixed 
holohedral  isometric  forms. 

The  fixed  forms  never  vary  in  the  slightest  degree 
in  appearance,  and  their  interfacial  angles  are  fixed 
quantities. 

A  Variable  Form:  A  variable  form  is  one  that  has 
one  or  more  variable  parameters  (m,  n,  or  p)  in  its 
general  symbol.  That  is,  the  symbol  contains  one 
or  more  parameters  to  which  various  values,  such 
as  2,  2J,  3,  etc.,  may  be  assigned  without  affecting 
the  naming  of  the  form.  The  trisoctahedron,  trape- 
zohedron,  hexoctahedron,  and  tetrahexahedron  are 
variable,  holohedral  isometric  forms. 

Two  or  more  variable  forms  of  the  same  name  may 
differ  considerably  in  shape  if  the  values  of  the  vari- 
able parameters  in  their 
symbols  are  materially 
different.  For  instance, 
Fig.  11  shows  two  trape- 
zohedrons  that  do  not 
resemble  each  other  very  F10-  H.  —  Trapezohedrons 

closely.     This  is  because    ^  s/mbols  <on left>  a:2a: 
,  T  ,    T      -    , ,  2a  and  (on  right)  a  :  3a  :  3a. 

the  symbol  of  the  one 

to  the  left  is  a  :  2a  :  2a,  while  that  of  the  one  to 
the  right  is  a  :  3a  :  3a. 

Interfacial  Angles  of  the  Fixed  Forms. 

Octahedron,  109°  28J'  (usually  given  as  109 J°). 
Dodecahedron,  60°,  90°,  and  120°. 
Cube,  90°. 


HOLOHEDRAL  DIVISION  25 

The  interfacial  angles  of  the  fixed  forms  are  often 
called  the  fixed  angles  of  a  system. 

Combination  of  Forms. 

It  will  be  found  that,  while  some  crystals  are 
bounded  by  a  single  crystal  form,  the  majority 
exhibit  more  than  one  such  form.  When  this  is  the 
case  the  shapes  of  the  faces  shown  in  Figs.  4  to  10, 
inclusive,  may  be  decidedly  changed,  but  their 
symbols  will  remain  unaltered. 

Determination  of  the  Number  of  Forms  on  a  Crystal. 

On  a  perfectly  developed  crystal  there  are  as 
many  forms  as  there  are  differently  shaped  and 
dimensioned  faces.  This  is  equivalent  to  saying 
that,  whether  unmodified  or  modified  by  the  pres- 
ence of  other  forms,  all  the  faces  of  a  given  crystal 
form  on  a  crystal  are  of  identically  the  same  shape 
and  size. 

Repetition  of  Forms  on  a  Crystal. 

Each  of  the  fixed  forms  can  occur  but  once  on  a 
crystal. 

Each  of  the  variable  forms  may  occur  an  indefinite 
number  of  times  on  a  crystal.  Theoretically,  one 
might  say  an  infinite  rather  than  an  indefinite 
number  of  times,  but,  practically,  the  number  of 
times  a  variable  form  is  repeated  on  a  crystal  is 
usually  small. 

In  order  that  a  given  variable  form  may  occur 
more  than  once  on  a  crystal,  it  is,  of  course,  necessary 
that  the  symbols  of  the  various  forms  of  the  same 
name  differ  as  regards  the  values  of  the  variable 


26 


ISOMETRIC  SYSTEM 


parameter  or  parameters.  Thus,  the  trapezohedron 
a  :  2a  :  2a  can  occur  but  once  on  a  crystal,  but  it 
may  be  combined  with  the  a  :  I J  a  :  1J  a  trapezo- 
hedron, the  a  :  3a  :  3a  trapezohedron,  etc. 

Holohedral   isometric   models   showing   repeated 
forms  are  difficult  to  obtain. 


edification  of  Fixed  Forms. 

It  will  be  found  useful  to  memorize  the  names  of 
the  forms  which  truncate  and  bevel  the  edges,  and 
truncate  the  corners,  of  each  of  the  fixed  forms,  as 
set  forth  in  the  following  table: 


Form  modi- 
fied. 

Form  truncat- 
ing edges. 

Form  beveling 
edges. 

Form  truncating 
corners. 

Octahedron 
Dodecahedron 
Cube 

dodecahedron 
trapezohedron 
dodecahedron 

trisoctahedron 
hexoctahedron 
tetrahexahedron 

cube 
cube  and  octahedron 
octahedron 

The  Triangle  of  Forms. 

It  will  be  found  advantageous  to  arrange  the 
crystal  forms  around  a  triangle  as  shown  in  Fig.  12. 
The  fixed  forms  are  at  the  corners  of  this  triangle, 
while  the  variable  ones  completely  fill  its  sides  and 
center.  Theoretically  there  is  an  infinite  number 
of  trisoctahedrons  along  the  left-hand  side,  each  of 
which  differs  from  the  others  as  regards  the  magni- 
tude of  the  variable  parameter.  Similarly  the  other 
sides  of  the  triangle  are  filled  with  infinite  numbers  of 
trapezohedrons  and  tetrahexahedrons,  while  the  in- 
terior of  the  figure  should  be  conceived  as  completely 
filled  with  an  infinite  quantity  of  hexoctahedrons. 


HOLOHEDRAL  DIVISION  27 

Utilization  of  the  Triangle  of  Forms. 

The  triangle  may  be  used  to  identify  small,  obscure 
forms  replacing  or  truncating  the  edges  between 
larger  and  more  easily  recognized  ones,  since  a  form 
lying  in  a  straight  line  between  two  other  forms  on 


a:a:ma  -*& —    a:a:coa      — ^»-  a:coa:ooa 

FIG.  12.  —  "  Triangle  of  Forms." 

the  triangle  will  replace  or  truncate  the  edge  between 
those  same  forms  on  a  crystal.  For  instance,  sup- 
pose a  small  face  replaces  the  edge  between  a  do- 
decahedron and  a  cube.  Reference  to  the  triangle 
shows  that  the  only  form  lying  in  a  straight  line 
between  the  dodecahedron  and  the  cube  is  a  tetra- 
hexahedron.  The  tetrahexahedron  must  then  be 
the  form  whose  name  is  sought.  From  what  was 
said  in  the  preceding  paragraph,  it  must  be  evident 
that  an  indefinite  number  (theoretically  infinite)  of 
tetrahexahedrons  may  replace  the  edge  between 
a  cube  and  dodecahedron.'  Similarly,  it  may  be 


28 


ISOMETRIC  SYSTEM 


ascertained  that  nothing  but  hexoctahedrons  can 
replace  the  edge  between  a  trapezohedron  and  a 
dodecahedron,  or  between  an  octahedron  and  a 
tetrahexahedron;  while  the  only  form  that  can 
replace  the  edge  between  two  trisoctahedrons  is 
another  trisoctahedron. 

Limiting  Forms  Defined. 

Limiting  forms  are  those  forms  which  a  variable 
form  approaches  in  appearance  (" shape")  as  the 
variable  parameter  (or  parameters)  in  its  symbol 
approaches  either  unity  or  infinity.  Thus,  a  trape- 
zohedron (a  :  ma  :  ma)  approaches  an  octahedron 
(a  :  a  :  a)  in  symbol  and  in  shape  as  m  approaches 
unity,  and  a  cube  (a  :  ooa  :  ooa)  as  m  approaches 
oo .  The  octahedron  and  cube  are,  then,  said  to  be 
limiting  forms  of  the  trapezohedron. 

Where  a  variable  form  is  situated  on  one  side  of 
the  triangle  of  forms,  the  two  forms  at  the  extremity 
of  that  side  are  its  limiting  forms.  The  hexocta- 
hedron,  in  the  interior  of  the  triangle,  has  all  the 
other  six  forms  as  its  limiting  forms. 


A  B 

FIG.  13.  —  Holohedral  isometric  crystals. 
A:  Octahedron  (o),  trapezohedron  (m),  trisoctahedron  (p), 
and  hexoctahedron  (x). 
B:  Cube  (a),  and  two  tetrahexahedrons  (h  and  /c). 


TETRAHEDRAL  HEMIHEDRAL  DIVISION    29 

TETRAHEDRAL   (INCLINED)   HEMIHEDRAL 
DIVISION 

Development  or  Derivation  of  the  Forms. 

Tetrahedral  hemihedral  isometric  forms  may  be 
conceived  to  be  developed  by  dividing  each  holo- 
hedral  form  by  means  of  the  three  principal  symmetry 
planes  into  eight  parts  (octants),  then  suppressing 
all  faces  lying  wholly  within  alternate  parts  thus 
obtained,  and  extending  all  the  remaining  faces  until 
they  meet  in  edges  or  corners. 

Symmetry. 

Tetrahedral  hemihedral  forms  possess  only  the  six 
secondary  symmetry  planes  characteristic  of  the 
isometric  system,  since  the  method  of  development 
outlined  necessarily  destroys  the  principal  symmetry 
planes. 

In  general,  it  may  be  said  that  the  planes  used  for 
dividing  holohedral  forms  in  the  development  of  hemi- 
hedral or  tetartohedral  forms  are  always  destroyed. 
/ 

'•Selection,  Position,   and  Designation  of  the  Crystal 
Axes. 

QThe  three  directions  used  as  crystal  axes  in  the 
holohedral  division  are  still  utilized  for  the  same 
purpose  in  the  tetrahedral  hemihedral  division?}  In 
other  words,  three  interchangeable  directions  at 
right  angles  to  each  other  are  used,  one  of  which  is 
held  vertically,  another  extending  horizontally  from 
right  to  left,  and  the  third  extending  horizontally 
from  front  to  back.  These  are,  however,  no  longer 


30 


ISOMETRIC  SYSTEM 


symmetry  axes,  as  they  were  in  the  holohedral  divi- 
sion. Each  is  called  an  a  axis,  as  in  the  holohedral 
division. 


Orienting  Crystals. 

Tetrahedral  hemihedral  crystals  are  oriented  by 
holding  a  symmetry  plane  (secondary)  so  that  it  ex- 
tends vertically  from  front  to  back,  then  rotating 
the  crystal  around  the  symmetry  axis  perpendicular 
to  this  plane  until  a  second  symmetry  plane  extends 
vertically  from  right  to  left,  and,  finally,  rotating 
the  crystal  around  a  vertical  axis  45°  either  to  the 
right  or  left.  When  this  has  been  done  the  sym- 
metry planes  will  occupy  their  proper  positions,  and 
the  crystal  axes  will  extend  in  the  proper  directions. 


Tetrahedral  Hemihedral  Isometric  Forms  Tabulated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

a:  a:  a 

4 

±  Trigonal     tristetrahedron  ) 

*       2 
a  :  ma  :  ma 

19 

(Fig.  15)           ...             ) 

^          2 

±  Tetragonal  tristetrahedron  ) 
(Fig.  16)                                   ) 

o  :  a  :  ma 

±~^— 

12 

trisoctahedron 

a  :  ma  :  na 

24 

hexoctahedron 

^          2 
a  :  ooo  :  ooa 

Dodecahedron  (Fig  6) 

2 
a:  a  :  ooa 

12 

hexahedron  (cube) 

2 

a  :  ma  :  <x>  a 

2 

TETRAHEDRAL  HEMIHEDRAL  DIVISION     31 


FIG.  14.  —  Positive  (on  left)  and  negative  (on  right)  tetra- 
hedrons containing  the  forms  from  which  they  are  derived. 
The  suppressed  faces  are  shaded. 


FIG.  15.  —  Positive  (on  left)  and  negative  (on  right)  trigonal 
tristetrahedrons  containing  the  forms  from  which  they  are 
derived.  The  suppressed  faces  are  shaded. 


FIG.  16.  —  Positive  (on  left)  and  negative  (on  right)  tetrago- 
nal tristetrahedrons  containing  the  forms  from  which  they 
are  derived.  The  suppressed  faces  are  shaded. 


Fia.  17.  —  Positive  (on  left)  and  negative  (on  right)  hextet- 
rahedrons  containing  the  forms  from  which  they  are  derived. 
The  suppressed  faces  are  shaded. 


32  ISOMETRIC  SYSTEM 

Synonyms  for  the  Names  of  the  Tetrahedral  Heinihe- 
dral  Isometric  Forms. 

Tetrahedron  —  none. 

Trigonal  tristetrahedron — tristetrahedron  or  hemi- 

tetragonal  trisoctahedron. 
Tetragonal  tristetrahedron — deltoid  dodecahedron 

or  hemitrigonal  trisoctahedron. 
Hextetrahedron  —  hemihexoctahedron. 

Symbols  of  Hemihedral  Forms. 

The  symbol  of  a  hemihedral  form  is  the  same  as 
that  of  the  holohedral  form  from  which  it  is  derived 
excepting  that  it  is  written  as  a  fraction  with  the 
figure  2  as  the  denominator.  This  does  not  mean 
that,  in  the  case  of  hemihedral  forms,  the  axes  are 
intersected  at  half  the  holohedral  axial  lengths,  but 
it  is  merely  a  conventional  method  of  indicating  that 
the  symbol  is  that  of  a  half  (hemihedral)  form.  The 

,    ,  a  :  a  :  a  .  n 

symbol  — •= is  read  a,  a,  a  over  2. 

Positive  (+)  and  Negative  (-)  Forms  Distinguished. 

All  those  forms  produced  by  the  suppression  of 
faces  lying  within  the  same  set  of  alternating  octants 
are  said  to  be  of  the  same  sign  (+  or  — ).  It  is 
customary  to  consider  those  forms  with  faces  largely 
or  entirely  included  within  the  upper  right  octant 
facing  the  observer  as  +,  while  those  with  faces  in 
the  upper  left  octant  are  — .  In  reality,  a  +  tetra- 
hedron differs  in  no  way  from  a  —  tetrahedron 
excepting  in  position;  and  a  tetrahedron  may  be 
held  in  either  the  -f  or  —  position  at  will.  It  is 
customary  to  hold  a  crystal  in  such  a  way  as  to  bring 
the  larger  and  more  prominent  faces  principally  or 


TETRAHEDRAL  HEMIHEDRAL  DIVISION     33 

entirely  into  the  upper  right  octant  facing  the 
observer,  which  will  make  these  forms  +  ones. 

If  the  sign  of  a  form  is  not  specifically  stated  to  be 
— ,  it  is  always  assumed  that  the  form  is  -f. 

The  forms  on  a  crystal  may  be  all  of  the  same  sign, 
or  +  and  —  forms  may  be  combined. 

If  a  -f-  and  a  —  form  of  the  same  name  and  with 
identical  parameters  are  equally  developed,  the 
combination  will  have  the  exact  external  shape  of  a 
holohedral  form.  Thus,  a  +  and  a  —  tetrahedron 
equally  developed  will  yield  an  octahedron.  On 
natural  crystals  equally  developed  +  and  —  forms 
often  differ  in  that  the  faces  of  one  may  be  brilliant, 
and  the  other  dull;  or  one  may  be  striated,  and  the 
other  unstriated  (see  p.  138);  or  one  may  striate 
another  form  (see  p.  139),  and  the  other  fail  to  do  so. 

Method  of  Determining  Tetrahedral  Hemihedral  Iso- 
metric Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  the  cube,  dodecahedron,  and  tet- 
rahexahedron  may  be  identified  easily  by  applying 
the  rules  already  given  for  the  determination  of 
holohedral  forms  of  the  same  name.  The  other  four 
forms  may  be  recognized  by  determining  the  symbol 
of  any  face  in  the  manner  described  in  the  discussion 
of  holohedral  forms,  dividing  this  symbol  by  2,  and 
then  ascertaining  from  the  table  the  name  of  the 
form  possessing  this  symbol. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  the  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 


34  ISOMETRIC  SYSTEM 

once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  Call  the  forms  +  or  —  according  to  the 
rules  already  set  forth.  It  is  assumed  that  the 
student  is  already  familiar  with  the  rules  for  recog- 
nizing those  forms  identical  in  shape  with  the  holo- 
hedral  ones  (see  p.  22). 

Tetrahedron:  A  single  face  in  the  center  of  the 
octant,  sloping  steeply  down  from  the  vertical  axis 
—  at  an  angle  of  54|°  with  the  horizontal. 

Trigonal  Tristetrahedron:  Three  faces  in  an  octant 
(although  not  necessarily  wholly  included  within  the 
same),  so  arranged  that  a  face  slopes  above  the 
center  of  the  octant  up  towards  the  vertical  axis 
at  an  angle  less  steep  than  is  shown  by  the  ^tetra- 
hedron. 

Tetragonal  Tristetrahedron:  Three  faces  lying  in  an 
octant  (although  not  necessarily  wholly  included 
therein),  so  arranged  that  in  the  unmodified  form  an 
edge  slopes  above  the  center  of  the  octant  up  toward 
the  vertical  axis.  Even  when  so  modified  that  the 
edge  is  lacking,  it  is  easy  to  see  that  two  faces 
extended  would  intersect  in  such  an  edge.  No  faces 
of  this  form  are  parallel  to  any  crystal  axis  or  to  each 
other.  This  will  serve  to  distinguish  this  form  from 
the  one  with  which  it  is  most  easily  confused,  namely, 
the  dodecahedron,  the  faces  of  which  are  arranged 
in  parallel  pairs. 

Hextetrahedron:  Six  faces  lying  in  an  octant  (al- 
though not  necessarily  wholly  included  therein), 
with  no  faces  parallel  to  a  crystal  axis  or  to  each 
other.  The  latter  portion  of  this  statement  will 
serve  to  distinguish  the  form  from  the  tetrahexa- 
hedron  with  which  it  may  be  most  easily  confused. 


TETRAHEDRAL  HEMIHEDRAL  DIVISION    35 

Distinction  Between  Hemihedral  and  Holohedral  Forms 
of  Ezactly  the  Same  Shapes. 

The  three  forms  last  named  in  the  table  of  tetra- 
hedral  hemihedral  forms  may  differ  in  no  way  what- 
ever from  the  corresponding  holohedral  forms  so  far 
as  external  shapes  are  concerned;  and  a  model  of  a 
cube,  for  instance,  may  with  equal  propriety  be  con- 
sidered either  a  holohedral  or  a  hemihedral  cube. 
On  natural  crystals,  however,  hemihedral  cubes, 
dodecahedrons,  and  tetrahexahedrons  differ  in 
molecular  structure  and  resulting  physical  proper- 
ties from  the  holohedral  forms  of  the  same  name. 
Holohedral  and  hemihedral  forms  of  the  same  name 
may  often  be  readily  distinguished  if  striated  by 
other  forms  (see  p.  139). 

Reason  Why  Some  Developed  or  Derived  Forms  do  not 
Differ  in  Shape  from  the  Forms  from  which  They 
Were  Derived. 

In  general,  it  is  true  that  a  derived  form  is  identical 
in  shape  with  the  form  from  which  it  was  developed 
when  no  faces  of  the  latter  lie  wholly  within  the  parts 
obtained  by  dividing  the  form  in  the  manner  specified 
in  the  rule  for  developing  the  derived  form. 

In  the  tetrahedral  division  of  the  isometric  system 
the  three  forms  which  fail  to  develop  into  others 
differing  from  the  holohedral  forms  in  shape  are  all 
characterized  by  the  fact  that  they  have  infinity 
in  their  symbol.  This  indicates  that  each  face  is 
parallel  to  one  or  two  of  the  axes,  and  must,  there- 
fore, lie  in  two  adjacent  octants.  Since  no  face  lies 
wholly  within  an  octant,  none  can  be  suppressed  in 
accordance  with  the  rule  given. 


36 


ISOMETRIC  SYSTEM 


Law  Governing  Combination  of  Forms. 

All  the  forms  on  any  one  crystal  must  possess  the 
same  degree  of  symmetry  as  regards  their  molecular 
structure. 

The  law  just  given  is  equivalent  to  the  statement 
that  it  is  impossible  to  have  holohedral  and  hemi- 
hedral,  or  holohedral  and  tetartohedral  forms  on  the 
same  crystal;  and  that  it  is  equally  impossible  for 
forms  belonging  to  different  hemihedral  or  tetarto- 
hedral divisions  to  occur  together. 

It  has  already  been  explained  that  some  forms 
really  hemihedral  so  far  as  their  internal  structure 
is  concerned  may  be  identical  with  holohedral  forms 
in  external  appearance.  These  may,  of  course,  be 
combined  with  the  other  hemihedral  forms  peculiar 
to  the  same  division  of  the  system.  The  same  state- 
ment applies  to  hemihedral  forms  which  are  identical 
in  appearance  with  hemihedral  forms  belonging  to 
other  divisions;  and  to  tetartohedral  forms  identical 
in  shape  with  those  in  other  tetartohedral  divisions 
or  with  hemihedral  or  holohedral  forms. 

Modification  of  Fixed  Forms. 


Form  modified. 

Form  truncating 
edges. 

Form  beveling 
edges. 

Form  truncating 
corners. 

+Tetrahedron  .  .  . 

cube 

+trigonal      tris- 

—tetrahedron 

tetrahedron 

—Tetrahedron  .  .  . 

cube 

—trigonal  tristet- 

+tetrahedron 

rahedron 

Cube  

dodecahedron 

tetrahexahedron 

rttetrahedron 

Dodecahedron  .  .  . 

itrigonal  tristetra- 

ihextetrahedron 

cube    and   ±tet- 

hedron 

rahedron 

PENTAGONAL  HEMIHEDRAL  DIVISION      37 

Miscellaneous. 

All  that  was  said  in  the  discussion  of  the  holohedral 
division  concerning  fixed  and  variable  forms,  inter- 
facial  angles  of  the  fixed  forms,  combination  of 
forms,  determination  of  the  number  of  forms,  repe- 
tition of  forms  on  a  crystal,  the  triangle  of  forms,  and 
limiting  forms  applies  with  equal  truth  to  all  hemi- 
hedral  and  tetartohedral  divisions,  excepting  that 
the  names  of  the  derived  forms  must  be  substituted 
in  these  statements  for  those  from  which  they  were 
derived. 


A  B 

FIG.  18.  —  Tetrahedral  hemihedral  isometric  crystals. 
A:   Cube  (a),  dodecahedron  (d),  +  and  —  tetrahedron  (o 
and  0i ),    +  hextetrahedron   (v},  and  —  trigonal  tristetrahe- 
dron  (ni). 

B:   +  tetrahedron  (o),  +  and  —  trigonal  tristetrahedron 
(n  and  ni),  and  dodecahedron  (d). 


PENTAGONAL  (PARALLEL)   HEMIHEDRAL 
DIVISION 

Development  or  Derivation  of  the  Forms. 

Pentagonal  hemihedral  isometric  forms  may  be 
conceived  to  be  developed  by  dividing  each  of  the 
holohedral  forms  by  means  of  the  six  secondary  sym- 
metry planes  into  twenty-four  parts,  then  suppress- 
ing all  faces  lying  wholly  within  alternate  parts  thus 


38 


ISOMETRIC  SYSTEM 


obtained,  and  extending  the  remaining  faces  until 
they  meet  in  edges  or  corners. 

Symmetry. 

Pentagonal  hemihedral  forms  possess  only  the 
three  principal  symmetry  planes  characteristic  of 
the  isometric  system. 

Selection,   Position,   and  Designation  of  the  Crystal 
Axes. 

The  three  directions  used  as  crystal  axes  in  the 
holohedral  and  tetrahedral  hemihedral  divisions  are 
still  utilized  for  the  same  purpose  in  the  pentagonal 
hemihedral  division.  In  other  words,  three  inter- 
changeable directions  at  right  angles  to  each  other 
are  used,  one  of  which  is  held  vertically,  another 
extending  horizontally  from  right  to  left,  and  the 
other  horizontally  from  front  to  back.  These  coin- 
cide in  position  with  the  principal  symmetry  axes, 
and  each  is  called  an  a  axis,  as  in  the  holohedral 
division. 

Orienting  Crystals. 

Pentagonal  hemihedral  crystals  are  oriented  in 
exactly  the  same  way  as  holohedral  ones  (see  p.  19). 


FIG.  19.  —  Pentagonal 
dodecahedron  containing  the 
form  from  which  it  is  derived. 
The  suppressed  faces  are 
shaded. 


FIG.  20.  —  Diploid  con- 
taining the  form  from  which 
it  is  derived.  The  suppressed 
faces  are  shaded. 


PENTAGONAL  HEMIHEDRAL  DIVISION      39 
Pentagonal  Hemihedral  Isometric  Forms  Tabulated. 


Name. 

Symbol. 

Number 
of  faces. 

Form  from  which 
derived. 

Pentagonal    dodecahedron  ) 
(Fig.  19)  ) 

a  :  ma  :  oo  a 
2 

12 

tetrahexahedron 

ninloirl  (fie   201 

a  :  ma  :  na 

24 

Octahedron  (Fig  4) 

2 
a:  a:  a 

3 

2 
a  :  a  :  oo  a 

10 

2 
a  :  oo  a  :  oo  a 

2 
a  :  ma  :  ma 

24 

2 
a:  a:  ma 

94. 

2 

Synonyms  for  the  Names  of  the  Pentagonal  Hemihe- 
dral Isometric  Forms. 

Pentagonal  dodecahedron  —  pyritohedron. 
Diploid  —  didodecahedron. 

Method  of  Determining  Pentagonal  Hemihedral  Iso- 
metric Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  the  octahedron,  dodecahedron, 
cube,  trapezohedron,  and  trisoctahedron  may  be 
identified  easily  by  applying  the  rules  already  given 
for  the  determination  of  holohedral  forms  of  the 
same  name.  The  other  two  forms  may  be  recog- 
nized by  determining  the  symbols  of  any  face  in  the 
manner  described  in  the  discussion  of  the  holohedral 
forms,  dividing  this  symbol  by  2,  and  then  ascertain- 
ing from  the  table  the  name  of  the  form  possessing 
this  symbol. 


40  ISOMETRIC  SYSTEM 

Suggestions  for  Attaining  Facility  in  the  Determina- 
tion of  the  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  It  is  unnecessary  to  distinguish  between 
positive  and  negative  forms  in  this  division.  It  is 
assumed  that  the  student  is  already  familiar  with 
the  rules  for  recognizing  those  forms  identical  in 
shape  with  the  holohedral  ones  (see  p.  22). 

Pentagonal  dodecahedron:  A  face  parallel  to  the 
right  and  left  axis,  and  sloping  down  toward  the 
observer  at  a  relatively  gentle  angle  —  less  than  45° 
from  the  horizontal.  The  angle  which  this  face 
makes  with  a  horizontal  plane  is  all  that  distinguishes 
it  from  the  face  of  a  dodecahedron  with  which  it 
may  most  easily  be  confused. 

If  after  orienting  a  crystal  a  prominent  face  is 
found  sloping  steeply  down  toward  the  observer  — 
over  45°  from  the  horizontal,  rotate  the  crystal  90° 
to  right  or  left  around  the  vertical  axis  before  apply- 
ing the  rule  just  given. 

Diploid:  Three  faces  lying  wholly  within  the 
octant,  so  arranged  that  none  of  the  edges  formed 
by  the  intersection  of  the  faces  in  the  unmodified 
form  point  directly  toward  the  vertical  axis.  This 
applies  either  to  edges  below  or  above  the  center  of 
the  octant.  It  is  useful  to  remember,  further,  that 
edges  formed  by  the  intersection  of  a  diploid  face 
with  an  octahedron  face,  either  below  or  above  the 
center  of  the  octant,  are  never  horizontal;  while  a 
trapezohedron  face  does  intersect  an  octahedron 
above  the  center  on  the  octant  in  a  horizontal  edge; 


PENTAGONAL  HEMIHEDRAL  DIVISION      41 

and  a  trisoctahedron  face  intersects  an  octahedron 
face  below  the  center  of  the  octant  in  a  horizontal 
edge.  It  should  be  noted  further  that  trapezohedron 
faces  intersect  cube  faces  in  edges  which  make  right- 
angles  with  each  other,  while  similar  edges  formed 
by  the  intersection  of  diploid  and  cube  faces  are 
never  at  right-angles.  These  statements  should 
serve  to  distinguish  the  diploid  from  either  the 
trapezohedron  or  trisoctahedron  with  which  it  is 
most  easily  confused. 


A  B 

FIG.  21.  —  Pentagonal  hemihedral  isometric  crystals. 
A:  Cube  (a),  pentagonal  dodecahedron  (e),  and  diploid  (s). 
B:  Cube  (a),  octahedron  (o),  and  pentagonal  dodecahedron 
(e). 

Application  to  the  Law   Governing   Combination   of 
Forms. 

The  law  already  stated  (see  p.  36)  governing  the 
combination  of  forms  should  not  be  forgotten  when 
naming  the  forms  on  a  pentagonal  hemihedral 
crystal.  One  of  the  commonest  mistakes  made  in 
determining  crystal  forms  is  to  mention  two  or  more 
forms  which  cannot  possibly  occur  together  on  a 
crystal,  as,  for  instance,  a  tetrahexahedron  and  a 
pentagonal  dodecahedron.  Any  form  in  the  pentag- 
onal hemihedral  division  may  be  combined  with 


42 


ISOMETRIC  SYSTEM 


any  other  form  in  the  same  division,  but  no  form  in 
the  pentagonal  hemihedral  division  may  be  com- 
bined with  any  form  in  the  tetrahedral  hemihedral 
or  holohedral  divisions  unless  that  form  occurs  also 
in  those  divisions.  The  student  is  very  apt  to  make 
the  mistake  mentioned  unless  he  can  reproduce  easily 
from  memory  the  following  table: 


Holohedral 
forms. 

Corresponding  tetrahedral 
hemihedral  forms. 

Corresponding  pentagonal 
hemihedral  forms. 

Octahedron  
Trapezohedron.  .  . 
Trisoctahedron... 
Hexoctahedron... 
Tetrahexahedron  . 

tetrahedron 
trigonal  tristetrahedron 
tetragonal  tristetrahedron 
hextetrahedron 
tetrahexahedron 

octahedron 
trapezohedron 
trisoctahedron 
diploid 
pentagonal  dodecahedron 

Dodecahedron  .  .  . 
Cube 

dodecahedron 
cube 

dodecahedron 
cube 

It  will  be  noted  from  the  above  table  that  the  cube 
and  dodecahedron  occur  in  all  three  of  the  divisions 
already  discussed,  and  may,  therefore,  be  combined 
with  any  other  forms  in  these  divisions. 

Further,  it  will  be  seen  that  the  octahedron, 
trapezohedron,  trisoctahedron,  and  tetrahexahedron 
occur  unchanged  in  name  or  shape  in  two  of  the 
divisions;  while  the  hexoctahedron  occurs  only  as 
a  holohedral  form. 


GTROIDAL  HEMIHEDRAL  DIVISION 

Gyroidal  hemihedral  isometric  forms  may  be  conceived 
to  be  developed  by  dividing  each  holohedral  form  by  both 
the  three  principal  and  the  six  secondary  symmetry  planes  into 
forty-eight  parts,  then  suppressing  all  faces  lying  wholly  within 


PENTAGONAL  TETARTOHEDRAL  DIVISION     43 

alternate  parts  thus  obtained,  and  extending  all  the  remain- 
ing faces  until  they  meet  in  edges  or  corners. 

As  the  hexoctahedron  is  the  only  isometric  form  with 
forty-eight  faces,  it  is  evident  that  a  hexoctahedron  face  is 
the  only  one  that  can  lie  wholly 
within  one  of  the  parts  obtained  by 
dividing  an  isometric  crystal  in  the 
manner  just  specified.  The  hex- 
octahedron  is,  then,  the  only  iso- 
metric form  from  which  a  gyroidal 

hemihedral  form  differing  from  the       ~ 

FIG.  22.  —  Pentagonal 
holohedral  one  in  shape  and  name  .      .,.  ,     ,    j  .   • 

i      j    •     j      mi  •  f        •     icositetrahedron  contain- 

can  be  derived     Tbs  new  form  ,s  form  from  which 

called  the  pentagonal  icositetrahe-  .,  .     ,    .  -,, 

j        /-n-     or»\      TO.-    t  .      it  ifl  derived.     The  sup- 

dron  (Fig.  22).    This  form  may  be 

...         .  ,,  ,  ...      j  j     .    .    pressed  faces  are  shaded, 

either  right-   or   left-handed,    but 

gyroidal  hemihedral  forms  are  so  rare  and  unimportant  that 
further  discussion  of  them  seems  unnecessary. 


PENTAGONAL  TETARTOHEDRAL  DIVISION 

Pentagonal  tetartohedral  isometric  forms  may  be  con- 
ceived to  be  developed  by  the  simultaneous  application  of 
any  two  hemihedrisms,  according  to 
the  principles  outlined  in  the  discus- 
sion of  the  trapezohedral  tetartohe- 
dral hexagonal  division  (see  p.  72). 

The  only  holohedral  form  which 
yields  a  tetartohedral  form  of  differ- 
ent shape  and  name  is  the  hexocta- 


FIG.  23.  —  Tetartoid 


hedron,  and  the  form  derived  from  containing  the  form 
it  is  called  a  tetartohedral  pen-  ^m  which  it  is  derived, 
tagonal  dodecahedron,  tetarto  hex-  The  suppressed  faces  are 
octahedron,  or  tetartoid  (Fig.  23).  snaded- 
Both  +  and  — ,  right-  and  left-handed  tetartoids  are  distin- 
guishable, but  tetartohedral  isometric  crystals  are  so  rare 
that  further  discussion  of  them  seems  unnecessary. 


44 


ISOMETRIC  SYSTEM 


Table  of  Isometric  Symbols  Used  by  Various  Authori- 
ties. 


Weiss. 

Naumann. 

Dana. 

Miller. 

Octahedron  

:  a  :  a 

o 

1 

(Ill) 

Trisoctahedron 

mO 

(Ml) 

Dodecahedron 

ooO 

(110) 

Trapezohedron  

:  ma  :  ma 

mOm 

m-m 

(hll) 

Hexahedron  (cube)  .... 
Hexoctahedron  
Tetrahexahedron  

:  ooa  :  ooa 
:  ma  :  na 

:  ma  :  <x>  a 

oo  O  oo 
mOn 
oo  On 

i-i 

m-n 

i-n 

(100) 
(hkl) 
(hkO) 

Weiss,  Naumann,  and  Dana  divide  the  holohedral  sym- 
bols by  2  and  by  4  when  referring  to  hemihedral  and  tetar- 
tohedral  forms,  respectively.  Miller  prefixes  various  Greek 
letters  when  forming  the  symbols  of  hemihedral  and  tetar- 
tohedral  forma. 


CHAPTER  III 
HEXAGONAL.    SYSTEM 

HOLOHEDRAL   DIVISION 
Symmetry. 

The  holohedral  division  of  the  hexagonal  system 
is  characterized  by  the  presence  of  one  principal  and 
six  secondary  symmetry  planes  which  lie  at  right 
angles  to  the  principal  symmetry  plane.  The 
secondary  symmetry  planes  are  arranged  in  two 
groups  each  of  which  contains  three  planes.  The 
planes  of  each  group  intersect  each  other  at  angles 
of  90°  and  120°,  and  are  interchangeable;  while  the 
planes  of  one  group  are  non-interchangeable  with 
those  of  the  other  group  which  they  intersect  at 
angles  of  30°,  90°,  or  150°. 

Selection,  Position,   and  Designation  of  the  Crystal 
Axes. 

The  principal  symmetry  axis  is  chosen  as  one  of 
the  crystal  axes,  is  held  vertically,  and  is  called  the 
c  axis.  Three  other  crystal  axes  are  so  selected  as 
to  coincide  with  either  group  of  interchangeable 
secondary  symmetry  axes.  One  is  held  horizontally 
from  right  to  left.  The  other  two  will,  then,  be 
horizontal,  but  will  make  angles  of  60°  with  each 
other  as  well  as  with  the  right  and  left  axis  (see 
Fig.  24).  Since  all  three  horizontal  axes  are  inter- 
changeable, they  are  all  called  a  axes.  None  of  the 

45 


46 


HEXAGONAL  SYSTEM 


horizontal  axes  is  interchangeable  with  the  vertical 
axis. 

This  is  the  only  system  in  which  more  than  three 
crystal  axes  are  used.  While  it  would  be  possible 
to  determine  the  holohedral  forms  by  the  use  of  three 


FlG.  24.  —  Crystal  axes  of  the  hexagonal  system. 

non-interchangeable  axes  intersecting  at  right  angles, 
as  in  the  orthorhombic  system  (see  p.  103),  this 
would  make  it  necessary  to  attach  different  names 
to  faces  identically  of  the  same  shape  and  size,  and 
would  in  no  way  suggest  the  six-  or  three-fold  arrange- 
ment of  faces  which  distinguishes  this  system.  It 
would  further  necessitate  the  devising  of  new  rules 
for  developing  hemihedral  and  tetartohedral  fomis; 
and  would  lead  to  so  many  difficulties  that  it  is  far 
simpler  to  use  the  three  interchangeable  horizontal 
axes  than  two  non-interchangeable  ones. 


HOLOHEDRAL  DIVISION  47 

Orienting  Crystals. 

Holohedral  hexagonal  forms  are  oriented  by  hold- 
ing the  principal  symmetry  plane  horizontally,  and 
either  set  of  interchangeable  secondary  symmetry 
planes  in  such  a  way  that  one  of  the  planes  will 
extend  vertically  from  right  to  left.  The  crystal  axes 
will  then  extend  in  the  proper  directions. 

The  Law  of  Rationality  or  Irrationality  of  Ratios  Be- 
tween Unit  Axial  Lengths. 

The  ratio  between  two  unit  axial  lengths  on  non- 
interchangeable  axes  is  always  an  irrational  quantity; 
while  the  ratio  between  the  unit  axial  lengths  on 
interchangeable  axes  is  not  only  a  rational  quantity, 
but  equals  unity. 

The  law  just  stated,  which  applies  to  all  systems, 
indicates  that  in  the  hexagonal  system  a  :  c  is  always 
an  irrational  quantity.  If  a  be  taken  as  unity, 
which  is  always  done,  c  may  be  greater  or  less  than 
unity,  but  is  always  an  irrational  quantity,  and  is 
usually  given  to  four  decimal  places.  As  an  illustra- 
tion, consider  the  hexagonal  mineral  beryl,  of  which 
emerald  is  a  variety.  The  ground-form  of  this  min- 
eral cuts  two  of  the  horizontal  and  the  vertical  axes 
at  such  distances  from  the  origin  as  will  make  the 
ratio  between  a  and  c  as  1  is  to  0.4989  (nearly).  The 
unit  axial  distances  of  this  mineral  are,  then,  a  =  1 
and  c  =  0.4989  (nearly).  The  value  of  c  differs  more 
or  less  for  all  hexagonal  minerals.  It  is,  then,  a  dis- 
tinguishing characteristic  of  each  hexagonal  mineral. 

Since  ra  and  n  are  always  rational  quantities  (see 
p.  15),  it  follows  that  na  :  c  and  a  :  me  are  irrational 
quantities;  while  a  :  na  is  a  rational  quantity. 


48 


HEXAGONAL  SYSTEM 


In  all  systems  but  the  isometric  m  may  be  less  than 
unity;  and  it  is  customary  to  apply  this  parameter 
(m)  to  the  unit  axial  length  of  the  vertical  axis,  n 
must  be  greater  than  unity  in  the  hexagonal  and  iso- 
metric systems  only. 

First  Order  Position  Defined. 

Forms  with  faces  whose  planes  cut  two  horizontal 
axes  equally  (at  equal  finite  distances  from  the 
origin),  and  are  parallel  to  the  third  horizontal  axis, 
are  said  to  be  in  the  first  order  position. 

Second  Order  Position  Defined. 

Forms  with  faces  whose  planes  cut  two  of  the 
horizontal  axes  equally  and  the  third  horizontal  axis 
at  a  distance  from  the  origin  which  is  half  that  cut 
off  on  the  other  two  horizontal  axes  are  said  to  be  in 
the  second  order  position. 

Third  Order  Position  Defined. 

Forms  with  faces  whose  planes  cut  all  three 
horizontal  axes  unequally  are  said  to  be  in  the  third 
order  position. 

Holohedral  Hexagonal  Forms  Tabulated. 


Name. 

Symbol. 

Number 
of  faces. 

1st  order  pyramid  (Fig.  25)  

a  :  a  :  oo  a  :  me 

12 

1st  order  priam  (Fig.  26)  

a  :  a  :  oo  a  :  oo  c 

6 

2nd  order  pyramid  (Fig.  27)  

2  a  :  a  :  2  a  :  me 

12 

2nd  order  prism  (Fig.  28)  

2  a  :  a  :  2  a  :  QO  c 

6 

Dihexagonal  pyramid  (Fig.  29)  

na  :  a  :  pa  :  me 

24 

Dihexagonal  prism  (Fig.  30)  

no  :  a  :  pa  :  oo  c 

12 

Basal-pinacoid  (Fig.  31)  

oo  a  :  oo  a  :  oo  o  :  c 

2 

HOLOHEDRAL  DIVISION 


49 


FIG.  25. —  1st  order  pyramid.       FIG.  26.  —  1st  order  prism. 


FIG.  27. — 2nd  order  pyramid.      FIG.  28.  —  2nd  order  prism. 


FIG.  29.  —  Dihexagonal 
pyramid. 


FIG.  30.  —  Dihexagonal 
prism. 


FIG.  31.  —  Basal  pinacoid. 


50  HEXAGONAL  SYSTEM 

Synonyms  for  the  Names  of  the  Holohedral  Hexagonal 
Forms. 

1st  order  pyramid  —  1st  order  bipyramid,  or  unit 

pyramid. 

1st  order  prism  —  unit  prism. 
2nd  order  pyramid  —  2nd  order  bipyramid. 
2nd  order  prism  —  none. 

Dihexagonal  pyramid  —  dihexagonal  bipyramid. 
Dihexagonal  prism  —  none. 
Basal-pinacoid  —  basal  plane. 

Method  of  Determining  Holohedral  Hexagonal  Forms 
by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  select  any  face  and  ascertain  the 
relative  distances  at  which  its  plane  intersects  the 
four  crystal  axes,  [remembering  that  no  face  or  faces 
extended  can  cut  the  vertical  axis  at  the  same 
distance  from  the  origin  as  it  cuts  any  of  the  hori- 
zontal axes^  If,  for  instance,  it  appears  that  the 
plane  of  the  face  selected  intersects  all  four  of  the 
axes,  but  that  the  three  horizontal  axes  are  all  cut 
at  unequal  distances  from  the  origin,  the  symbol  of 
that  face  (and  of  the  form  of  which  it  is  a  part)  is 
na  :  a  :  pa  :  me.  By  referring  to  the  table  of  holo- 
hedral  hexagonal  forms  (which  should  be  memorized 
as  soon  as  possible)  it  is  seen  that  the  form  is  the 
dihexagonal  pyramid.  If  more  than  one  form  is 
represented  on  the  crystal,  each  may  be  determined 
in  the  same  way. 

The  parameter  p  in  the  symbol  of  the  dihexagonal 
pyramid  and  prism  is  not  an  independent  variable, 

79 

but  is,  in  fact,  equal  to  — —r.    When  n  equals  3, 


HOLOHEDRAL  DIVISION  51 

for  instance,  p  will  equal  f  or  1 J.     It  might  be  better 
always  to  use instead  of  p,  but,  if  the  equality 

fl  J- 

of  the  two  symbols  is  always  borne  in  mind,  no 
confusion  need  result. 

It  is  necessary  to  write  the  symbol  of  hexagonal 
forms  in  such  a  way  as  will  make  the  second  part  of 
each  symbol  always  a  (or  ooa  in  the  case  of  the 
basal-pinacoid).  Thus,  na  :  a  :  pa  :  me  is  correct, 
while  na  :  pa  :  a  :  me  is  incorrect ;  and  a  :  2  a  : 
2  a  :  me  is  not  the  symbol  of  the  2nd  order  pyramid, 
while  2  a  :  a  :  2  a  :  me  is  the  correct  symbol  of  this 
form. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  faces  of  different  shape  or  size 
seen. 

1st  order  pyramid:  A  face  sloping  down  from  the 
vertical  axis  directly  towards  the  observer. 

1st  order  prism:  A  vertical  face  extending  from 
right  to  left. 

2nd  order  pyramid:  A  face  sloping  down  from  the 
vertical  axis  directly  to  the  right  or  left. 

A  2nd  order  pyramid  differs  in  no  way  from  a  1st 
order  pyramid  excepting  in  position  with  respect  to 
the  horizontal  crystal  axes;  and  a  twelve-faced 
pyramid  may  be  placed  in  either  the  1st  or  2nd 
order  position  at  will.  Such  a  pyramid  may,  then, 
be  considered  either  a  1st  or  a  2nd  order  pyramid 
depending  upon  the  set  of  interchangeable  symmetry 


52  HEXAGONAL  SYSTEM 

axes  with  which  the  crystal  axes  are  chosen  to  coin- 
cide. It  is  only  when  forms  in  both  the  1st  and 
2nd  order  position  occur  on  the  same  crystal  that 
it  is  necessary  to  distinguish  between  1st  and  2nd 
order  pyramids. 

2nd  order  prism:  A  vertical  face  extending  from 
front  to  back. 

As  is  the  case  with  the  2nd  order  pyramid,  a  2nd 
order  prism  differs  in  no  way  from  a  1st  order  prism 


FIG.  32.  —  Diagrams  showing  relations  of  the  1st  order 
(A),  2nd  order  (B),  and  dihexagonal  (C)  pyramids  and  prisms 
to  the  horizontal  crystal  axes. 

excepting  in  position  with  respect  to  the  horizontal 
crystal  axes;  and  all  that  was  said  in  the  preceding 
section  relative  to  the  2nd  order  pyramid  applies 
with  equal  truth  to  the  2nd  order  prism. 

It  is  customary  to  select  the  horizontal  crystal 
axis  in  such  a  way  as  will  place  the  largest  and  most 
prominent  twelve-faced  pyramid  or  six-faced  prism 
in  the  1st  order  position. 

Pyramids  and  prisms  intersecting  in  horizontal 
edges  are  always  of  the  same  order. 

Dihexagonal  pyramid:  A  face  sloping  down  from 
the  vertical  axis  in  such  a  way  that  its  plane  inter- 
sects all  three  horizontal  crystal  axes  at  unequal 
finite  distances  from  the  origin. 


HOLOHEDRAL  DIVISION 


53 


Dihexagonal  prism:  A  vertical  face  whose  plane 
intersects  all  thjee  horizontal  crystal  axes  at  unequal 
finite  distances  from  the  origin. 

Basal-pinacoid:  A  horizontal  face  on  top  of  a 
crystal. 

Fixed  and  Variable  Forms. 

The  only  fixed  holohedral  hexagonal  forms  are  the 
1st  and  2nd  order  prisms  and  the  basal-pinacoid, 
and  forms  derived  therefrom. 

Fixed  Angles  of  the  Hexagonal  System. 

The  only  fixed  angles  in  this  system  are  those 
between  the  fixed  forms  just  mentioned,  namely, 
90°,  120°  (or  60°),  and  150°  (or  30°). 

Miscellaneous. 

The  general  statements  made  in  the  discussion 
of  the  holohedral  division  of  the  isometric  system 


FIG.  33.  —  Holohedral  hexagonal  crystals. 

A:  Basal-pinacoid  (c),  1st  order  pyramid  (r),  and  2nd  order 
prism  (a). 

B:  Basal-pinacoid  (c),  1st  order  prism  (a),  two  1st  order 
pyramids  (p  and  u),  2nd  order  pyramid  (r),  and  dihexagonal 
pyramid  (v). 

regarding  combination  of  forms,  determination  of 
the  number  of  forms,  repetition  of  forms  on  a  crystal, 


54  HEXAGONAL  SYSTEM 

and  limiting  forms  applies  with  equal  truth  to  all 
the  divisions  of  the  hexagonal  system.  It  may  be 
mentioned,  however,  that  repetitions  of  the  same 
variable  form  are  very  common  in  the  hexagonal 
system,  and  crystal  models  showing  such  repeated 
forms  are  not  difficult  to  obtain. 

RHOMBOHEDRAL   HEMIHEDRAL   DIVISION 
Development  or  Derivation  of  the  Forms. 

Rhombohedral  hemihedral  hexagonal  forms  may 
be  conceived  to  be  developed  by  dividing  each  of  the 
holohedral  forms  by  means  of  the  principal  symmetry 
plane  and  the  set  of  interchangeable  secondary  sym- 
metry planes  containing  the  crystal  axes  into  twelve 
parts,  or  dodecants,  then  suppressing  all  faces  lying 
wholly  within  alternate  parts  thus  obtained,  and 
extending  the  remaining  faces  until  they  meet  in 
edges  or  corners. 

Symmetry. 

Rhombohedral  hemihedral  forms  possess  three 
interchangeable  secondary  symmetry  planes  which 
intersect  along  a  common  line  at  angles  of  60°  or  120°. 

Selection,  Position,   and  Designation  of  the  Crystal 
Axes. 

The  three  directions  used  as  crystal  axes  in  the 
holohedral  division  are  still  utilized  for  the  same 
purpose  in  the  rhombohedral  hemihedral  division. 
In  other  words,  the  vertical  or  c  axis  lies  at  the  inter- 
section of  the  three  secondary  symmetry  planes; 
while  the  three  interchangeable  crystallographic 
directions  used  as  horizontal  or  a  axes  are  so  held 


RHOMBOHEDRAL  HEMIHEDRAL  DIVISION     55 


that  one  of  them  extends  from  right  to  left,  and  each 
of  them  bisects  an  angle  between  two  of  the  second- 
ary symmetry  planes. 

Orienting  Crystals. 

Rhombohedral  hemihedral  forms  are  oriented  by 
holding  a  symmetry  plane  vertically  from  front  to 
back,  then  rotating  the  crystal  around  the  axis  per- 
pendicular to  this  plane  until  two  other  symmetry 
planes  making  angles  of  60°  to  120°  with  the  plane 
first  mentioned  are  held  vertically.  The  crystal 
axes  will  then  extend  in  the  proper  directions. 

Rhombohedral  Hemihedral  Hexagonal  Forms  Tabu- 
lated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

o  :  a  :  oo  a  :  me 

Hexagonal  scalenohedron  ) 

2 

na  :  a  :  pa  :  me 

fdihexagonal     pyra- 

(Fig  35)                            ) 

2 

\    mid 

a  :  a  :  ooo  :  ooc 

2 
2  a  :  a  :  2  a  :  me 

2nd  order  pyramid  (Fig.  27)  . 

2 
2a  :  a  :  2a  :ooc 

2 

na:a:pa:<x>c 

19 

2 
oo  a  :  oo  a  :  oo  a  :  c 

2 

Synonyms  for  the  Names  of  the  Rhombohedral  Hemi- 
hedral Hexagonal  Forms. 

Rhombohedron  —  none. 
Scalenohedron  —  none. 


56 


HEXAGONAL  SYSTEM 


Positive  and  Negative  Forms  Distinguished. 

All  those  forms  produced  by  the  suppression  of 
faces  lying  within  the  same  set  of  alternating  dode- 
cants  are  said  to  be  of  the  same  sign  (+  or  — ).  It 
is  customary  to  consider  those  forms  with  faces 


FIG.  35.  —  Hexagonal 
scalenohedron  contain- 
ing the  form  from  which 
it  is  derived.  The  sup- 


FIG.  34.  —  Positive  (on  left)  and 
negative  (on  right)  rhombohedrons 
containing  the  forms  from  which 
they  are  derived.  The  suppressed 
faces  are  shaded. 


largely  or  entirely  included  within  the  upper  dode- 
cant  directly  facing  the  observer  as  +,  while  those 
with  faces  in  the  upper  dodecant.at  the  back  of 
the  crystal  furthest  from  the  observer  are  — .  In 
reality  a  +  rhombohedron  differs  in  no  way  from 
a  —  rhombohedron  excepting  in  position;  and  a 
rhombohedron  may  be  held  in  either  the  +  or  — 
position  at  will.  It  is  customary  to  hold  a  crystal 
in  such  a  way  as  to  bring  the  largest  and  most 
prominent  rhombohedron  face  principally  or  entirely 
into  the  upper  dodecant  facing  the  observer,  which 
will  make  this  form  a  +  one.  It  is  possible,  but 
unnecessary,  to  distinguish  between  +  and  —  hex- 
agonal scalenohedrons. 


RHOMBOHEDRAL  HEMIHEDRAL  DIVISION     57 

If  the  sign  of  a  form  is  not  specifically  stated  as 
being  — ,  it  is  always  assumed  that  the  form  is  +. 

The  forms  on  a  crystal  may  all  be  of  the  same 
sign,  or  +  and  —  forms  may  be  combined. 

Method   of   Determining    Rhomb ohedral  Hemihedral 
Hexagonal  Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  all  the  forms 
but  the  rhombohedron  and  scalenohedron  may  be 
easily  identified  by  applying  the  rules  already  given 
for  the  determination  of  holohedral  forms  of  the  same 
name.  The  two  hemihedral  forms  new  in  shape 
may  be  recognized  by  determining  the  symbol  of  any 
face  in  the  manner  described  in  the  discussion  of 
holohedral  forms,  dividing  the  symbol  by  2,  and 
then  ascertaining  from  the  table  the  name  of  the 
form  possessing  this  symbol. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the  fol- 
lowing descriptions  (which  should  be  learned  at  once) 
apply  to  the  face  or  faces  of  different  shape  or  size 
seen.  It  is  assumed  that  the  student  is  already  fa- 
miliar with  the  rules  for  recognizing  those  forms  iden- 
tical in  shape  with  the  holohedral  ones  (see  p.  51). 

+  Rhombohedron:  A  face  sloping  down  from  the 
vertical  axis  directly  toward  the  observer.  A  rhom- 
bohedron has  three  faces  at  each  end  of  the  vertical 
axis  so  arranged  that  a  face  on  top  is  directly  above 
an  edge  below. 

—  Rhombohedron:  A  face  sloping  down  from  the 
vertical  axis  directly  away  from  the  observer,  at  the 
back  of  the  crystal. 


58  HEXAGONAL  SYSTEM 

Hexagonal  scalenohedron:  A  face  sloping  down 
from  the  vertical  axis  in  such  a  way  that  its  plane 
intersects  all  three  horizontal  crystal  axes  at  unequal 
finite  distances  from  the  origin. 

The  hexagonal  scalenohedron  is  most  readily 
confused  with  the  2nd  order  pyramid.  To  dis- 
tinguish them,  it  should  be  remembered  that  the 
upper  and  lower  faces  of  the  latter  always  intersect 
in  horizontal  edges,  and  that  the  interfacial  angles 
of  the  2nd  order  pyramid,  measured  across  edges 
converging  towards  the  vertical  axis,  are  all  equal. 
Neither  statement  is  true  as  regards  the  hexagonal 
scalenohedron. 

Rules  and  Conventions  Relating  to  Rhombohedrons. 

As  may  be  gathered  from  the  statements  already 
made  in  this  volume  (see  p.  13),  the  unit  rhombo- 
hedron  in  the  case  of  any  given  mineral  species  is 
usually  the  rhombohedron  occurring  most  commonly 
on  crystals  of  that  mineral.  In  the  case  of  rhombo- 
hedral  minerals  with  a  well-developed  rhombohedral 
cleavage  (see  p.  142),  however,  it  is  sometimes  found 
more  convenient  to  select  the  cleavage  rhombohedron 
as  the  unit  rhombohedron,  and  to  call  all  distances 
at  which  this  unit  rhombohedron  cuts  the  a  and  c 
axes  the  unit  axial  distances  a  and  c.  a  is  made 
equal  to  unity,  and  c  is  then  some  irrational  quantity 
either  greater  or  less  than  unity.  It  is  customary 
to  designate  the  unit  rhombohedron  by  the  symbol 
R  which  may  be  -f-  or  —  according  to  its  position  on 
the  crystal. 

All  other  rhombohedrons  than  R  will  cut  the  a 
and  c  axes  at  such  distances  that  if  a  is  made  equal 


RHOMBOHEDRAL  HEMIHEDRAL  DIVISION     59 

to  unity,  me  will  be  some  rational  multiple  of  c.  If 
ra  is  equal  to  2,  the  rhombohedron  is  represented  by 
the  symbol  2  R,  either  -f  or  — ;  while  if  m  is  equal 
to  |,  the  rhombohedron  is  represented  by  the  symbol 
\  R,  either  +  or  — .  Similarly, 
a  rhombohedron  intersecting 
the  vertical  axis  at  3c  may 
be  represented  by  the  symbol 
3  R,  etc. 

It  may  be  readily  proved 
geometrically  (although  to  FlG.  36~  iew  from 
offer  such  a  proof  is  beyond  above  of  a  unit  rhombo- 
the  scope  of  this  work)  that  hedron,  with  its  edges 
any  rhombohedron  which  truncated  by  -  \R,  and 

truncates  the  edges  of  another    the  edge*  uof  th(f  *atter 

0  .„    .    .  truncated  by  +  \  R. 

rhombohedron   will  intersect 

the  vertical  axis  at  one-half  the  c  value  of  the  trun- 
cated rhombohedron  and  will  be  of  opposite  sign. 
In  other  words,  +  R  may  have  its  edges  truncated 
by  —  \  R;  —  \  R  may  have  its  edges  truncated  by 
+  \  R,  etc.  Stated  differently,  —  \  R  will  truncate 
the  edges  of  +  R;  or  —  2  R  will  truncate  the  edges 
of  +  4  R. 

From  what  has  been  said  it  is  evident  that  a  + 
rhombohedron  always  truncates  a  —  rhombohedron 
or  vice  versa.  In  order  to  ascertain  whether  one 
rhombohedron  is  truncating  another,  it  is  only  neces- 
sary to  determine  whether  one  rhombohedron  inter- 
sects the  other  in  parallel  edges.  If  a  rhombohedron 
is  intersected  by  other  rhombohedron  faces  of 
opposite  sign  so  as  to  produce  parallel  edges,  the 
former  is  truncating  the  latter.  Fig.  36  illustrates  a 
crystal  viewed  from  above  that  shows  +  R,  —  \  R, 


GO  HEXAGONAL  SYSTEM 


and  -f  \  R.  If  the  rhombohedron  represented  in 
the  case  just  mentioned  as  —  \  R  be  taken  as  the 
unit  rhombohedron,  +  R,  the  other  rhombohedrons 
shown  will  be  —  2  R  and  —  |  R,  respectively. 


B 

FIG.  37.  —  Rhombohedral  hemihedral  hexagonal  crystals. 
A :  Two   +  rhombohedrons  (r  and  M)  and  two  scaleno- 
hedrons  (y  and  v) . 

B:    -j-  rhombohedron  (r)  and  two  scalenohedrons  (w  and  v). 

PYRAMIDAL  HEMIHEDRAL  DIVISION 

Development  or  Derivation  of  the  Forms. 

Pyramidal  hemihedral  hexagonal  forms  may  be 
conceived  to  be  developed  by  dividing  each  of  the 
holohedral  forms  by  means  of  all  six  secondary 
symmetry  planes  into  twelve  parts,  then  suppressing 
all  faces  lying  wholly  within  alternate  parts  thus 
obtained,  and  extending  the  remaining  faces  until 
they  meet  in  edges  or  corners. 

Symmetry. 

Pyramidal  hemihedral  forms  possess  only  one 
symmetry  plane  which  is  in  the  position  of  the 
principal  symmetry  plane  existing  in  the  holohedral 
division.  It  is,  however,  in  the  pyramidal  hemi- 
hedral division  a  secondary  rather  than  a  principal 


PYRAMIDAL  HEMIHEDRAL  DIVISION       61 


symmetry  plane  since  there  are  no  interchangeable 
symmetry  planes  perpendicular  to  it. 

Pyramidal  hemihedral  hexagonal  forms  are,  then, 
characterized  by  the  presence  of  one  secondary 
symmetry  plane,  and  a  general  six-fold  arrangement 
of  faces. 


Selection,   Position, 
Axes. 


and  Designation  of  the   Crystal 


The  vertical  or  c  crystal  axis  is  made  to  coincide 
with  the  secondary  symmetry  axis.  Three  inter- 
changeable horizontal  axes  parallel  to  prominent 
crystallographic  directions  at  angles  of  60°  or  120° 
to  each  other  are  also  selected,  and  one  of  these  is  so 
placed  as  to  extend  from  right  to  left.  Being  inter- 
changeable, all  are  called  a  axes. 


FIG.  38.  —  3rd  order  pyra- 
mid containing  the  form  from 
which  it  is  derived.  Sup- 
pressed faces  are  shaded. 


FIG.  39.  —  3rd  order  prism 
containing  the  form  from 
which  it  is  derived.  Sup- 
pressed faces  are  shaded. 


Orienting  Crystals. 

The  secondary  symmetry  plane  is  held  horizon- 
tally. The  crystal  is  then  rotated  around  the  sym- 
metry axis  until  the  most  prominent  pyramid  or 
prism  lies  in  the  first  order  position.  The  crystal 
axes  will  then  extend  in  the  proper  directions. 


62  HEXAGONAL  SYSTEM 

Pyramidal  Hemihedral  Hexagonal  Forms  Tabulated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

3rd  order  pyramid  (Fig.  38)  . 
3rd  order  prism  (Fig.  39)  .... 

1st  order  pyramid  (Fig.  25)  . 
1st  order  prism  (Fig.  26)  
2nd  order  pyramid  (Fig.  27) 
2nd  order  prism  (Fig.  28)  ... 
Basal-pinacoid  (Fig.  31)  

na  :  a  :  pa  :  me 

12 
6 

12 
6 
12 
6 
2 

dihexagonal  pyramid 
dihexagonal  prism 

1st  order  pyramid 
1st  order  prism 
2nd  order  pyramid 
2nd  order  prism 
basal-pinacoid 

2 
na  :  a  :  pa  :  oo  c 

2 

a  :  a  :  oo  g  ;  me 

2 
o  :  a  :  ooo  :  ooc 

2 

2  a  :  a  :  2  a  :  me 

2 

2a  :  o  :  2o  :  ooc 

2 

oo  a  :  oo  a  :  oo  a  :  c 

2 

Synonyms  for  the  Names  of  the  Pyramidal  Hemihedral 
Hexagonal  Forms. 

3rd  order  pyramid  —  3rd  order  bipyramid. 
3rd  order  prism  —  none. 

Method  of  Determining  Pyramidal  Hemihedral  Hex- 
agonal Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  all  the  forms  but  the  3rd  order 
pyramid  and  prism  may  be  identified  easily  by 
applying  the  rules  already  given  for  the  determina- 
tion of  holohedral  forms  of  the  same  name.  3rd 
order  pyramids  and  prisms  may  be  recognized  by 
determining  the  symbol  of  any  face  in  the  manner 
already  described  in  the  discussion  of  holohedral 
forms,  dividing  this  symbol  by  2,  and  then  ascertain- 
ing from  the  table  the  name  of  the  form  possessing 
this  symbol. 


PYRAMIDAL  HEMIHEDRAL  DIVISION       63 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  It  is  possible,  but  unnecessary,  to  dis- 
tinguish between  +  and  —  forms  in  this  division. 
It  is  assumed  that  the  student  is  already  familiar 
with  the  rules  for  recognizing  those  forms  identical 
in  shape  and  position  with  the  holohedral  ones  (see 
p.  51). 

3rd  order  pyramid:  A  face  sloping  down  from  the 
vertical  axis  so  that  its  plane  intersects  all  three 
horizontal  crystal  axes  at  un- 
equal  finite  distances  from  the 
origin. 

The  3rd  order  pyramid  dif- 
fers in  no  way  from  the  1st  or 
2nd  order  pyramid  excepting 
in  position  with  respect  to  the 
horizontal  crystal  axes.  All 

three  types  of  12-faced  pyra-    ,FlG'  «,- Diagram 

.  .  .  showing  the  relations  of 

mids  may  have  the  same  ap-  the   lst   order  (dotted 

pearance;  and  any  such  pyra-  lines),  2nd  order  (broken 
mid  may  be  held  at  will  as  a  lines),  and  3rd  order 
1st,  2nd,  or  3rd  order  pyra-  (solid  Unes)  Pyramids 

mid.  The  3rd  order  pyramid  and  Prisms  to  the  hori- 
.  .  \J  zontal  crystal  axes, 

is  skewed  or  twisted  through 

a  small  angle  (less  than  30°)  either  to  the  right  or 
left  away  from  the  position  of  the  1st  or  2nd  order 
pyramid.  Fig.  40  shows  how  the  horizontal  axes  are 
cut  by  1st,  2nd,  and  3rd  order  pyramids  and  prisms. 
3rd  order  prism:  A  vertical  face  whose  plane  inter- 


64 


HEXAGONAL  SYSTEM 


sects  the  three  horizontal  crystal  axes  at  unequal 
finite  distances  from  the  origin. 

All  that  was  said  in  the  preceding  section  relative 
to  the  3rd  order  pyramid  applies  with  equal  truth 
to  the  3rd  order  prism. 


FIG.  41.  —  Pyramidal  hemihedral  hexagonal  crystals. 

A:  Basal-pinacoid  (c),  1st  order  prism  (m),  two  1st  order 
pyramids  (x  and  y),  2nd  order  pyramid  (s),  and  3rd  order 
prism  (h). 

B:  Basal-pinacoid  (c),  2nd  order  prism  (a),  and  3rd  order 
pyramid  (u). 

TRIGONAL  HEMIHEDRAL   DIVISION 
Development  or  Derivation  of  the  Forms. 

Trigonal  hemihedral  hexagonal  forms  may  be  con- 
ceived to  be  developed  by  dividing  each  of  the  holo- 
hedral  forms  by  means  of  the  three  secondary  sym- 
metry planes  containing  the  horizontal  crystal  axes 
into  six  parts,  then  suppressing  all  faces  lying  wholly 
within  alternate  parts  thus  obtained,  and  extending 
the  remaining  faces  until  they  meet  in  edges  or 
corners. 

Symmetry. 

The  trigonal  hemihedral  division  of  the  hexagonal 
system  is  characterized  by  the  presence  of  one 
principal  and  three  interchangeable  secondary  sym- 
metry planes  which  lie  at  right  angles  to  the  principal 


TRIGONAL  HEMIHEDRAL  DIVISION         65 


symmetry  plane.     The  interchangeable  symmetry 
planes  make  angles  of  60°  or  120°  with  each  other. 

Selection,   Position,   and  Designation  of  the  Crystal 
Axes. 

The  principal  symmetry  axis  is  chosen  as  the 
vertical  or  c  crystal  axis ;  while  three  interchangeable 
horizontal  directions,  each  of  which  bisects  the  angle 
between  two  secondary  symmetry  planes,  constitute 
the  horizontal  or  a  axes.  One  of  these  is  so  held  as 
to  extend  from  right  to  left. 

Orienting  Crystals. 

Trigonal  hemihedral  forms  are  oriented  by  holding 
the  principal  symmetry  plane  horizontally,  and  one 
secondary  symmetry  plane  vertically  and  extending 
from  front  to  back.  The  crystal  axes  will  then 
extend  in  the  proper  direction. 

Trigonal  Hemihedral  Hexagonal  Forms  Tabulated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

±  1st  order  trigonal  pyra-) 

a  :  a  :  oo  a  :  me 

mid  (Fig.  42)  ) 
±  1st  order  trigonal  prism  ( 

2 

o  :  a  :  ooo  :  ooc 

(Fig.  43)                          f 

2 

Ditrigonal    pyramid    (Fig.  ( 

na  '.  a  :  pa  :  me 

44)  J 

2 
na  :  a  :  pa  :  QO  c 

2 

2  a  :  a  :  2  a  :  me 

12 

2 
2a  :  a  :  2a  :  ooc 

2 
oo  a  :  oo  a  :  oo  a  :  c 

2 

66 


HEXAGONAL  SYSTEM 


FIG.  42.  —  Positive  (on  left)  and  negative  (on  fight)  1st 
order  trigonal  pyramids  containing  the  forms  from  which 
they  are  derived.  The  suppressed  faces  are  shaded. 


FIG.  43.  —  Positive  (on  left)  and  negative  (on  right)  1st 
order  trigonal  prisms  containing  the  forms  from  which  they 
are  derived.  The  suppressed  faces  are  shaded. 


FIG.  44.  —  Ditrigonal  pyra- 
mid containing  the  form  from 
which  it  is  derived.  The  sup- 
pressed faces  are  shaded. 


FIG.  45. —  Ditrigonal  prism 
containing  the  form  from 
which  it  is  derived.  The  sup- 
pressed faces  are  shaded. 


TRIGONAL  HEMIHEDRAL  DIVISION         67 

Synonyms  for  the  Names  of  the  Trigonal  Hemihedral 
Hexagonal  Forms. 

1st  order  trigonal  pyramid  —  trigonal  bipyramid 

of  the  1st  order. 
Ditrigonal  pyramid  —  di trigonal  bipyramid. 

Positive  and  Negative  Forms  Distinguished. 

All  those  forms  produced  by  the  suppression  of 
faces  lying  within  the  same  set  of  alternating  dode- 
cants  are  saM  to  be  of  the  same  sign  (-£  or  — ).  It 
is  possible,  but  unnecessary,  to  distinguish  between 
+  and  —  ditrigonal  pyramids  and  prisms.  It  is 
customary  to  consider  a  trigonal  prism  or  pyramid 
with  a  face  or  faces  extending  from  right  to  left 
between  the  vertical  axis  and  the  observer  as  +, 
while  one  with  such  a  face  or  faces  back  of  the 
vertical  axis  is  — .  In  reality,  a  +  trigonal  pyramid 
differs  in  no  way  from  a  —  trigonal  pyramid  except- 
ing in  position;  and  a  trigonal  pyramid  may  be  held 
in  either  the  +  or  —  position  at  will.  The  same 
statements  hold  as  regards  the  trigonal  prism. 
Convention  requires  that  the  largest  and  most 
prominent  trigonal  pyramid  or  prism  should  be  held 
in  such  a  way  as  to  bring  it  into  the  +  p'osition. 

All  the  other  statements  regarding  +  and  —  forms 
made  in  the  discussion  of  the  rhombohedral  hemi- 
hedral  division  (see  p.  57)  apply  with  equal  truth 
to  the. division  under  consideration. 

Method  of  Determining  Trigonal  Hemihedral  Hexag- 
onal Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  the  2nd  order  pyramid  and  prism 


68  HEXAGONAL  SYSTEM 

and  the  basal-pinacoid  may  be  identified  easily  by 
applying  the  rules  already  given  for  the  determina- 
tion of  holohedral  forms  of  the  same  name. 

The  four  hemihedral  forms  differing  in  shape  from 
the  holohedral  ones  from  which  they  were  derived 
may  be  recognized  by  determining  the  symbol  of  any 
face  in  the  manner  described  in  the  discussion  of 
holohedral  forms,  dividing  this  symbol  by  2,  and 
then  ascertaining  from  the  table  the  name  of  the 
form  possessing  this  symbol. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  It  is  assumed  that  the  student  is  already 
familiar  with  the  rules  for  recognizing  those  forms 
identical  in  shape  with  the  holohedral  ones  (see 
p.  51). 

+  1st  order  trigonal  pyramid:  A  face  sloping  down 
from  the  vertical  axis  directly  toward  the  observer. 
This  face  occupies  exactly  the  same  position  as  that 
of  a  +  rhombohedron.  However,  a  1st  order  trig- 
onal pyramid  differs  from  a  rhombohedron  in  that 
the  three  faces  at  one  end  of  the  vertical  axis  inter- 
sect those  at  the  other  end  in  edges  which  are 
horizontal. 

—  1st  order  trigonal  pyramid:  A  face  sloping  down 
from  the  vertical  axis  directly  away  from  the  ob- 
server at  the  back  of  the  crystal.  This  face  occupies 
exactly  the  same  position  as  that  of  a  —  rhombo- 
hedron. 


TRIGONAL  HEMIHEDRAL  DIVISION         69 

+  1st  order  trigonal  prism:  A  vertical  face  extend- 
ing directly  from  right  to  left  between  the  vertical 
axis  and  the  observer. 

—  1st  order  trigonal  prism:  A  vertical  face  extend- 
ing directly  from  right  to  left  at  the  back  of  the 
crystal. 

Ditrigonal  pyramid:  A  face 
sloping  down  from  the  vertical 
axis  in  such  a  way  that  its 
plane  intersects  the  three  hori- 
zontal crystal  axes  at  unequal 
finite  distances  from  the  origin. 
The  six  faces  at  each  end  of  the  8howing  the  relation  of 
vertical  axis  occupy  exactly  the  the  faces  of  the  trig- 
same  positions  as  the  six  faces  onal  hemihedral  di- 
making  up  half  of  a  scalenohe-  tri.g°nal  P^amid  and 
dron,  but  may  be  distinguished  P™  to  the  honzonta. 
e  crystal  axes.  (Com- 

from  scalenohedron  faces  by  the  pare  ^^  j^g  54 ) 

fact  that  the  faces  at  opposite 

ends  of  the  vertical  axis  intersect  in  edges  that  are 

horizontal. 

Ditrigonal  prism:  A  vertical  face  whose  plane 
intersects  all  three  horizontal  crystal  axes 'at  unequal 
finite  distances  from  the  origin. 

Hemimorphism. 

A  hemimorphic  crystal,  as  already  stated  (p.  17), 
is  one  in  which  the  law  of  axes  (see  p.  16)  is  violated 
so  far  as  one  crystal  axis  is  concerned.  In  other 
words,  on  a  hemimorphic  crystal  the  opposite  ends 
of  one  crystal  axis  are  not  cut  by  the  same  number 
of  similarly  placed  faces.  For  instance,  there  may 
be  one  or  more  pyramids  on  one  end  of  a  crystal  axis, 


70  HEXAGONAL  SYSTEM 

and  only  a  basal  pinacoid  on  the  other;  or  the  forms 
at  both  ends  of  an  axis  may  have  the  same  names, 
but  different  slopes. 

Theoretically,  hemimorphic  forms  may  occur  in 
all  divisions  of  the  hexagonal  system,  but  they  are 
relatively  unimportant  on  any  kind  of  crystals 
already  discussed  excepting  trigonal  hemihedral 
hexagonal  ones. 

Naming  Hemimorphic  Forms. 

It  is  customary  to  hold  the  axis  whose  ends  are 
treated  differently  vertically.  After  properly  orient- 
ing the  crystal  the  forms  on  the  upper  end  of  the 
crystal  are  given  first,  then  the  crystal  is  turned  up- 
side down,  and  those  on  the  other  end  are  named. 
Forms  common  to  both  ends,  like  prisms  and  pina- 
coids  (other  than  the  basal  pinacoid),  are  mentioned 
but  once. 

In  writing  out  the  names  of  the  forms  on  a  hemi- 
morphic crystal  it  is  customary  to  separate  the  names 
of  the  forms  on  the  differing  ends  of  the  crystal  by 
means  of  a  horizontal  line. 

Importance  of  Hemimorphism  in  the  Trigonal  Hemi- 
hedral Hexagonal  Division. 

With  one  possible  exception,  all  natural  minerals 
crystallizing  in  this  division  of  the  hexagonal  system 
are  hemimorphic,  that  is,  the  opposite  ends  of  their 
vertical  axes  are  not  intersected  by  the  same  number 
of  similar  faces  similarly  placed.  This  eliminates 
the  principal  symmetry  plane,  and  gives  the  trigonal 
and  ditrigonal  pyramids  the  appearance  of  rhom- 
bohedrons  and  scalenohedrons.  That  the  crystals 


TRIGONAL  HEMIHEDRAL  DIVISION 


71 


resulting  are  not  rhombohedral  is,  however,  usually 
shown  plainly  by  the  presence  of  a  prominent  trig- 
onal prism,  a  form  which  does  not  occur  in  the 
rhombohedral  hemihedral  division.  This  gives  trig- 


A  B 

FIG.  47.  —  Typical  horizontal  sections  of  the  trigonal 
hemihedral  mineral  tourmaline. 

onal  hemihedral  crystals  cross-sections  which  are 
either  triangular  or  (more  commonly)  spherical- 
triangular.  Fig.  47  shows  two  typical  cross-sections 
of  the  mineral  tourmaline  which  is  the  commonest 
species  crystallizing  in  this  division  of  the  system. 


A  B 

FIG.  48.  —  Trigonal  hemihedral  hexagonal  crystals 

(hemimorphic). 

A:  +  and  —1st  order  trigonal  pyramids  (r  and  o),  —1st 
order  trigonal  prism  (mi},  and  2nd  order  prism  (a).  On  other 
end:  +  and  —1st  order  trigonal  pyramids  (r  and  e). 

B:  -f  and  —1st  order  trigonal  pyramids  (r  and  o),  +  and 
—  1st  order  trigonal  prisms  (m  and  mi),  ditrigonal  pyramid 
(u),  ditrigonal  prism  (h),  and  2nd  order  prism  (a).  On  other 
end:  +  and  —1st  order  trigonal  pyramids  (r  and  o). 


72 


HEXAGONAL  SYSTEM 


TRAPEZOHEDRAL    HEMIHEDRAL   DIVISION 

Trapezohedral  hemihedral  forms  may  be  conceived  to  be 
developed  by  dividing  each  holohedral  form  by  the  principal 
and  all  the  secondary  symmetry 
planes  into  24  parts,  then  suppress- 
ing all  faces  lying  wholly  within 
alternate  parts  thus  obtained,  and 
extending  all  the  remaining  faces 
until  they  meet  in  edges  or  corners. 

As  the  dihexagonal  pyramid  is 
the  only  hexagonal  form  with  24 
faces,  it  is  evident  that  a  dihexa- 
gonal pyramid  face  is  the  only  one  FIG.  49.  —  Hexagonal 
that  can  lie  whoUy  within  one  of  trapezohedron  contain- 
the  parts  obtained  by  dividing  a  ing  the  form  from  which 
hexagonal  crystal  in  the  manner  ft  js  derived.  The  sup- 
just  specified.  The  dihexagonal  pressed  faces  are  shaded, 
pyramid  is,  then,  the  only  hexa- 
gonal form  from  which  a  trapezohedral  hemihedral  form  differ- 
ing from  the  holohedral  one  in  shape  and  name  can  be  derived. 
This  new  form  is  called  the  hexagonal  trapezohedron  (Fig.  49). 
This  form  may  be  either  right  or  left-handed,  but,  since  no 
mineral  is  known  to  crystallize  in  this  division,  its  further 
discussion  seems  unnecessary. 


TRAPEZOHEDRAL   TETARTOHEDRAL   DIVISION 
Development  or  Derivation  of  the  Forms. 

Trapezohedral  tetartohedral  hexagonal  forms  may 
be  conceived  to  be  developed  by  the  simultaneous 
application  of  the  rhombohedral  and  trapezohedral 
hemihedrisms.  In  other  words,  a  holohedral  form 
is  first  divided  into  dodecants  by  means  of  the 
principal  symmetry  plane  and  the  three  secondary 
symmetry  planes  containing  the  crystal  axes,  as  in 
the  development  of  rhombohedral  hemihedral  forms ; 
and  faces  or  portions  of  faces  lying  within  alter- 


TRAPEZOHEDRAL  TETARTOHEDRAL 


73 


nating  dodecants  are  marked  tentatively  as  sub- 
ject to  suppression.  The  holohedral  form  is  then 
divided  by  means  of  the  principal  and  all  of  the 
secondary  symmetry  planes  into  twenty-four  parts, 
as  in  the  development  of  trapezohedral  hemihedral 
forms;  and  faces  or  portions  of  faces  lying  within 
alternating  parts  thus  obtained  are  marked  tenta- 
tively as  subject  to  suppression.  If,  after  this  has 
been  done,  it  is  found  that  any  crystal  face  has  been 
marked  in  such  a  way  as  to  indicate  that  all  portions 
of  it  are  tentatively  subject  to  suppression,  that 


Upper  six  faces  as 
viewed  from  above. 


Lower  six  faces  j 
viewed  from  below. 


FIG.  50.  —  Diagram  to  illustrate  the  development  of  the  2nd 
order  trigonal  pyramid  from  the  2nd  order  pyramid  as  ex- 
plained in  the  text. 

crystal  face  is  suppressed;  but,  if  all  or  any  portion 
of  a  crystal  face  remains  unmarked  as  subject  to 
suppression,  that  face  is  extended  until  it  meets 
other  similar  faces  in  edges  or  corners. 

As  an  illustration  of  the  process  just  outlined, 
consider  a  2nd  order  pyramid  (see  Fig.  50).  If  this 
form  is  divided  by  means  of  the  principal  symmetry 


74  HEXAGONAL  SYSTEM 

plane  and  the  secondary  symmetry  planes  contain- 
ing the  crystal  axes,  and  faces  or  parts  of  faces  lying 
within  alternating  dodecants  thus  obtained  are 
marked  tentatively  as  subject  to  suppression,  parts 
of  faces  1  and  2,  5  and  6,  and  9  and  10  on  top  of 
the  crystal;  and  3  and  4,  7  and  8,  and  11  and  12  on 
the  other  end  should  be  so  marked,  as  indicated 
by  the  vertically  hatched  portions  on  Fig.  50.  If, 
then,  the  form  be  divided 
by  means  of  the  principal 
symmetry  plane  and  all 
six  secondary  symmetry 
planes  into  twenty-four 
parts,  and  faces  or  parts 
of  faces  lying  within  alter- 
nating parts  thus  obtained  FlG  51  _  2nd  order  trig. 
be  marked  tentatively  as  onal  pyramid  containing  the 
subject  to  suppression,  form  from  which  it  is  de- 
the  parts  of  the  faces  so  rived-  Suppressed  faces  are 
marked  will  be  those  num- 
bered 1,  3,  5,  7,  9,  and  11  on  top  of  the  crystal,  and  2, 
4,  6,  8,  10,  and  12  on  the  bottom,  as  indicated  by  the 
horizontally  hatched  portions  in  Fig.  50.  This  leaves 
the  half-faces  4,  8,  and  12,  on  top,  and  the  half-faces 
1,  5,  and  9,  on  the  other  end  of  the  crystal  unhatched, 
while  all  the  remaining  faces  are  completely  hatched. 
If,  then,  we  extend  the  faces  partially  unhatched  until 
they  meet  in  edges  or  corners  as  illustrated  by  Fig. 
51,  we  shall  obtain  the  trapezohedral  tetartohedral 
derivative  of  the  second  order  pyramid,  namely,  the 
2nd  order  trigonal  pyramid.  By  applying  this 
method  to  the  other  holohedral  forms,  their  trapezo- 
hedral tetartohedral  derivatives  may  be  ascertained. 


TRAPEZOHEDRAL   TETARTOHEDRAL          75 

Symmetry. 

Trapezohedral  tetartohedral  forms  possess  no 
symmetry  planes  whatever,  but  show  a  three-fold 
or  six-fold  arrangement  of  faces. 

Selection,   Position,   and  Designation  of  the  Crystal 
Axes. 

The  four  directions  used  as  crystal  axes  in  the 
holohedral  division  are  still  utilized  for  the  same 
purpose  in  the  trapezohedral  tetartohedral  division. 
In  other  words,  one  vertical  or  c  axis  and  three  inter- 
changeable horizontal  or  a  axes  intersecting  at  angles 
of  60°  or  120°  are  utilized.  One  of  the  latter  is  held 
from  right  to  left. 

Orienting  Crystals. 

The  absence  of  all  symmetry  planes,  and  the 
presence  of  two  sets  of  prominent  interchangeable 
crystallographic  directions  which  may  be  so  placed 
as  to  occupy  the  position  of  the  horizontal  crystal 
axes  makes  it  impossible  to  give  any  rules  for  orient- 
ing trapezohedral  tetartohedral  crystals  based  en- 
tirely on  symmetry  planes  or  crystallographic 
directions. 

Since  quartz  is  the  only  mineral  which  occurs  at  all 
commonly  in  recognizable  trapezohedral  tetartohe- 
dral crystals,  it  seems  best  to  suggest  rules  for  orienta- 
tion applicable  especially  to  that  mineral.  These 
are  as  follows: 

On  most  crystals  one  crystallographic  direction 
emerging  on  the  surface  of  the  crystal  at  a  corner 
formed  by  the  intersection  of  three  or  six  faces 
making  equal  angles  with  each  other  is  usually  very 


76 


HEXAGONAL  SYSTEM 


prominent.  This  is  selected  as  the  vertical  or  c 
axis,  and  is  not  interchangeable  with  any  other 
crystallographic  direction.  The  crystal  is  then 
rotated  around  the  vertical  axis  until  prominent 
prism  faces  occupy  the  1st  order  position;  or,  if  a 
prominent  prism  is  lacking,  prominent  pyramidal 
faces  are  placed  in  the  1st  order  position.  The  three 
interchangeable  horizontal  crystal  axes  will  then 
extend  in  the  proper  directions. 

Trapezohedral  Tetartohedral  Hexagonal  Forms  Tabu- 
lated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

a  :  a  :  oo  a  :  me 

±2nd  order  trigonal  pyra-( 

4 
2  a  :  a  :  2  a  :  me 

mid  (Fig.  51)                   \ 

4 

±2nd  order  trigonal  prism  ( 

2a:a:  2a:  <x>c 

(Fig.  52)  J 
Trigonal     trapezohedron     1 

4 
na  :  a  :  pa  :  me 

(Fig.  53)  ( 

4 
na  :  a  :  pa  :  oo  c 

4 

a  :  a  :  oo  a  :  we 

4 
oo  a  :  oo  a  :  oo  a  :  c 

4 

Synonyms  for  the  Names  of  the  Trapezohedral  Te- 
tartohedral Hexagonal  Forms. 

2nd  order  trigonal  pyramid  —  trigonal  bipyramid 
of  the  2nd  order. 

2nd  order  trigonal  prism  —  unsymmetrical  trig- 
onal prism. 


TRAPEZOHEDRAL  TETARTOHEDRAL    77 

Trigonal  trapezohedron  —  quadrilateral  trapezo- 
hedron. 

Positive  and  Negative  Forms  Distinguished. 

It  is  possible  to  distinguish  between  +  and  - 
variations  of  each  of  the  trapezohedral  tetartohedral 
forms  which  differ  in  shape  from  the  holohedral  ones 


FIG.  52.  —  Diagrams  showing  relations  of  the  positive  (on 
left)  and  negative  (on  right)  2nd  order  trigonal  pyramid  and 
prism  to  the  horizontal  crystal  axes. 


FIG.  53.  —  Trigonal  trape- 
zohedron containing  the  form 
from  which  it  is  derived.  The 
suppressed  faces  are  shaded. 


FIG.  54.  —  Diagram  show- 
ing the  relation  of  the  tetartc- 
hedral  ditrigonal  prism  to  the 
horizontal  crystal  axes. 


from  which  they  are  derived.  It  is,  however, 
unnecessary  to  differentiate  between  +  and  —  trig- 
onal trapezohedrons  and  ditrigonal  prisms. 

+   and   —   rhombohedrons  are  distinguished  in 
exactly  the  same  way  as  are  +  and  —  rhombohe- 


78  HEXAGONAL  SYSTEM 

drons  in  the  rhombohedral  hemihedral  division  (see 
p.  56). 

It  is  customary  to  call  those  trigonal  prisms  and 
pyramids  -f  which  have  a  face  or  faces  extending 
directly  from  front  to  back  at  the  right  of  the  vertical 
axes;  while  those  with  a  similar  face  or  faces  at  the 
left  of  the  vertical  axis  are  called  —  .  See  Fig.  52. 

Symbols  of  Tetartohedral  Forms. 

The  symbol  of  a  tetartohedral  form  in  any  system 
is  the  same  as  that  of  the  holohedral  form  from  which 
it  is  derived  excepting  that  it  is  written  as  a  fraction 
with  the  figure  4  as  the  denominator.  This  does 
not  mean  that,  in  the  case  of  the  tetartohedral  forms, 
the  axes  are  intersected  at  one-fourth  the  holohedral 
axial  lengths,  but  is  merely  a  conventional  method 
of  indicating  that  the  symbol  is  that  of  a  quarter 


/.  ,      i         i        i\      r  mi  11 

(tetartohedral)  form.     The  symbol 
is  read  na,  a,  pa,  me  over  4. 


Method  of  Determining  Trapezohedral  Tetartohedral 
Hexagonal  Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  the  1st  order 
prism  and  basal-pinacoid  may  be  identified  easily  by 
applying  the  rules  already  given  for  the  determina- 
tion of  holohedral  forms  of  the  same  name.  The 
tetartohedral  forms  new  in  shape  may  be  recognized 
by  determining  the  symbol  of  any  face  in  the  manner 
described  in  the  discussion  of  holohedral  forms, 
dividing  this  symbol  by  4,  and  then  ascertaining 
from  the  table  the  name  of  the  form  possessing  this 
symbol. 


TRAPEZOHEDRAL  TETARTOHEDRAL    79 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  It  is  assumed  that  the  student  is  already 
familiar  with  the  rules  for  recognizing  those  forms 
identical  in  shape  with  the  holohedral  ones  (see 
p.  51). 

d=  Rhombohedron:  Since  the  trapezohedral  tetar- 
tohedral  rhombohedrons  differ  in  no  way  excepting 
in  internal  molecular  arrangement  from  the  rhombo- 
hedral  hemihedral  forms  of  the  same  name,  rules  for 
the  recognition  of  the  +  and  —  rhombohedrons  given 
in  the  description  of  rhombohedral  hemihedral  forms 
(see  p.  57)  may  be  used  in  identifying  such  forms 
in  this  division. 

+  2nd  order  trigonal  pyramid:  A  face  which 
slopes  down  from  the  vertical  axis  directly  to  the 
right. 

The  2nd  order  trigonal  pyramid  does  not  differ  in 
appearance  from  the  1st  order  trigonal  pyramid 
occurring  in  the  trigonal  hemihedral  division,  but 
does  differ  from  the  latter  in  its  position  with  refer- 
ence to  the  horizontal  crystal  axes. 

—  2nd  order  trigonal  pyramid:  A  face  which  slopes 
down  from  the  vertical  axis  directly  to  the  left. 

+  2nd  order  trigonal  prism:  A  vertical  face  extend- 
ing from  front  to  back  at  the  right  of  the  vertical 
axis. 

•  The  2nd  order  trigonal  prism  does  not  differ  in 
appearance  from  the  1st  order  trigonal  prism  occur- 
ring in  the  trigonal  hemihedral  division,  but  does 


80  HEXAGONAL  SYSTEM 

differ  from  the  latter  in  its  position  with  reference  to 
the  horizontal  crystal  axes. 

—  2nd  order  trigonal  prism:  A  vertical  face  extend- 
ing from  front  to  back  at  the  left  of  the  vertical  axis. 

Trigonal  trapezohedron:  A  face  sloping  down  from 
the  vertical  axis  in  such  a  way  that  its  plane  inter- 
sects all  three  horizontal  crystal  axes  at  unequal 
finite  distances  from  the  origin. 

It  is  possible,  but  not  necessary,  to  distinguish 
between  right-  and  left-handed  trigonal  trapezo- 
hedrons. 

The  trigonal  trapezohedron,  like  the  1st  and  2nd 
order  trigonal  pyramids  and  the  rhombohedron, 
has  three  faces  at  each  end  of  the  vertical  axis,  but 
the  faces  on  top  do  not  intersect  those  below  in 
horizontal  edges,  as  is  the  case  with  the  trigonal 
pyramid;  nor  is  a  face  on  top  directly  above  an  edge 
below,  as  is  the  case  with  the  rhombohedron.  The 
three  faces  on  one  end  appear,  in  fact,  to  have  been 
twisted  around  the  vertical  axis  through  a  small 
angle  (less  than  30°)  to  the  right  or  left,  placing  them 
in  an  unsymmetrical  position  with  reference  to  those 
at  the  other  end  of  the  crystal. 

The  trigonal  trapezohedron  is  most  apt  to  be 
confused  with  a  2nd  order  trigonal  pyramid.  These 
may  usually  be  distinguished  easily  if  the  following 
tests  are  applied: 

If  a  vertical  plane  is  passed  through  an  edge  be- 
tween two  equally  steep  1st  order  faces,  it  will  bisect 
the  angle  between  two  diverging  edges  of  a  2nd  order 
trigonal  pyramid  face  directly  above  or  below  the 
edge  first  mentioned.  Equally  steep  1st  order  faces 
in  this  division  of  the  system  are  those  of  the  1st 


TRAPEZOHEDRAL  TETARTOHEDRAL    81 

order  prism,  and,  in  the  case  of  quartz,  those  between 
the  most  prominent  +  and  —  rhombohedrons. 

The  statement  just  made  will  not  be  found  true 
where  trigonal  trapezohedrons  he  over  or  under  the 
edges  formed  by  the  intersection  of  equally  steep  1st 
order  faces. 

Ditrigonal  prism:  The  tetartohedral  ditrigonal 
prism  closely  resembles  the  trigonal  hemihedral  form 
of  the  same  name  (see  p.  66),  but  differs  therefrom 
in  that  it  appears  to  have  a  symmetry  plane  running 
from  right  to  left  (see  Fig.  54),  while  the  hemihedral 
form  has  a  symmetry  plane  extending  from  front  to 
back  (see  Fig.  46). 

General  Observations. 

It  has  already  been  mentioned  that  quartz  is  the 
only  common  mineral  crystallizing  in  this  division 
of  the  hexagonal  system,  and  it  should  be  noted  that 
the  crystallization  of  quartz  is  peculiar  in  that  the 
most  prominent  faces  are. a  1st  order  prism  (some- 
times missing)  and  an  equally  or  unequally  devel- 
oped +  and  —  rhombohedron  the  faces  of  which 
make  equal  angles  with  the  prism  faces.  In  the 
absence  of  other  forms,  the  combination  last  men- 
tioned appears  to  be  rhombohedral  hemihedral, 
while  the  first  combination  appears  to  be  holohedral. 
The  presence  of  either  a  trigonal  trapezohedron  or 
trigonal  pyramid  is  sufficient  to  prove  the  crystal 
tetartohedral,  but  these  forms  are  very  rare.  The 
trigonal  and  ditrigonal  prisms  and  basal-pinacoid 
are  almost  never  found  on  quartz  crystals. 


82  HEXAGONAL  SYSTEM 


A 
FIG.  55.  —  Trapezohedral  tetartohedral  hexagonal  crystals. 

A:  +  and  —  rhombohedrons  (r  and  z),  1st  order  prism 
(w),  —  2nd  order  trigonal  pyramid  (s),  and  trigonal  trapezo- 
hedron  (x). 

B:  +  and  —  rhombohedron  (r  and  z)  equally  developed, 
1st  order  prism  (m),  and  —  2nd  order  trigonal  pyramid  (s). 

Application  of  the  Law  Governing  Combination  of 
Forms. 

It  has  already  been  mentioned  that  one  of  the 
commonest  mistakes  made  in  determining  crystal 
forms  is  the  naming  of  two  or  more  forms  which 
cannot  possibly  occur  on  the  same  crystal  since  their 
presence  would  be  in  direct  violation  of  the  law 
governing  the  combination  of  forms  (see  p.  36). 
The  student  should  thoroughly  familiarize  himself 
with  the  table  on  p.  83  if  he  wishes  to  avoid  the 
mistake  mentioned. 

Inspection  of  this  table  will  show  that  the 
basal-pinacoid  is  the  only  form  that  occurs  un- 
changed in  all  five  of  the  divisions  tabulated.  It  is, 
then,  the  only  form  that  can  be  combined  with  any 
other  of  the  forms  in  the  divisions  considered. 
Further,  it  should  be  noticed  that  the  1st  order 
prism  occurs  unchanged  in  all  divisions  but  the 
trigonal  hemihedral,  while  the  2nd  order  pyramid 
and  prism  occur  unchanged  in  all  divisions  but  the 
trapezohedral  tetartohedral.  The  forms  in  the  3rd 


TRAPEZOHEDRAL   TETARTOHEDRAL 


83 


ii  is 

9  x  *r  a 

5  f*| 

s-    -fe  *»    0 


•M    >M      1 

I  S  1 1  i 


2 


fiil 

Q)      ~*A      i*^        £ 

1.1  II 


. 
ll 


!!!!! 

ro   co   a   a  ^ 

~*    T-t    M    (M    80 


ft    ^ 

I  1  is 

*rti 


u 


84 


HEXAGONAL  SYSTEM 


order  position  are  apt  to  give  the  most  trouble, 
since  the  dihexagonal  pyramid  yields  a  form  of  new 
name  and  shape  in  each  of  the  hemihedral  and 
tetartohedral  divisions;  while  the  dihexagonal 
prism  yields  a  form  of  new  name  and  shape  in  all 
divisions  but  the  rhombohedral  hemihedral. 


RHOMBOHEDRAL  TETARTOHEDRAL  DIVISION 

Rhombohedral  tetartohedral  forms  may  be  conceived  to 
be  developed  by  the  simultaneous  application  of  the  rhom- 
bohedral and  pyramidal  hemihedrisms,  according  to  the  prin- 
ciples outlined  in  the  discussion  of  the  trapezohedral  tetar- 
tohedral hexagonal  division  (see  p.  72).  The  names  of  the 
resulting  rhombohedral  tetartohedral  forms  together  with 
their  symbols,  number  of  faces,  and  the  name  of  the  corre- 
sponding holohedral  forms  are  shown  on  the  following  table: 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

a  :  a  :  oo  a  :  me 

4 
2  a  :  a  :  2  a  :  me 

4 
na  :  a  :  pa  :  me 

4 
na  :  a  :  pa  :  oo  c 

g 

4 

a  :  a  :  ooa  :  ooc 

4 

2a:  a  :  2a  :  ooc 

5 

4 

oo  a  :  oo  a  :  oo  a  :  c 

2 

4 

The  rhombohedral  tetartohedral  3rd  order  prism  is  exactly 
like  the  form  of  the  same  name  in  the  pyramidal  hemihedral 


RHOMBOHEDRAL  TETARTOHEDRAL 


85 


division  (see  p.  63).  The  1st  order  rhombohedron  is  exactly 
like  the  rhombohedron  occurring  in  the  rhombohedral  hemi- 
hedral  and  the  trapezohedral  tetartohedral  divisions;  while 
the  2nd  and  3rd  order  rhombohedrons  differ  from  the  1st 
order  form  of  the  same  name  only  in  position  with  reference 
to  the  crystal  axes.  The  former  is,  of  course,  in  the  2nd 
order  position,  while  the  latter  is  in  the  3rd  order  position. 

Further  consideration  of  this  division  seems  unnecessary 
since  few  minerals  are  rhombohedral  tetartohedral,  and  these 
are  comparatively  rare. 

Table  of  Hexagonal  Symbols  Used  by  Various  Authori- 
ties. 


Weiss. 

Nau- 
mann. 

Dana. 

Miller. 

1st  order  pyramid  
1st  order  prism  .... 

a  :  a:  c 
a  :  a  :  c 

oa:  me 
o  a  :  oo  c 

mP 
ooP 

m 
I 

(ftOAt) 
(1010) 

2nd  order  pyramid  

2a:  a 

2  a  :  me 

mP2 

m-2 

(M  2  h  2  i) 

2nd  order  prism  .... 

2a:  a 

2  a  :  oo  c 

ooP2 

i-2 

(1020) 

Dihexagonal  pyramid  
Dihexagonal  prism  
Basal-pinacoid  

na:  a: 
na:  a: 
oo  a  :  c 

pa  :  me 
pa  :  oo  c 
o  a  :  oo  a  :  c 

mPn 
oo  Pn 
OP 

m-n 
i-n 

o 

(hkli) 
(hklO) 
(0001) 

Weiss,  Naumann,  and  Dana  divide  the  holohedral  sym- 
bols by  2  and  by  4  when  referring  to  hemihedral  and  tetarto- 
hedral forms,  respectively.  Miller  prefixes  various  Greek 
letters  when  forming  the  symbols  of  hemihedral  and  tetarto- 
hedral forms. 


CHAPTER   IV 
TETRAGONAL  SYSTEM 

HOLOHEDRAL  DIVISION 

Symmetry. 

The  holohedral  division  of  the  tetragonal  system 
is  characterized  by  the  presence  of  one  principal  and 
four  secondary  symmetry  planes  which  lie  at  right 
angles  to  the  principal  symmetry  plane.  The 
secondary  symmetry  pjanes  are  arranged  in  two 
pairs.  The  planes  of  each 
pair  intersect  each  other  at 
an  angle  of  90°  and  are 
interchangeable;  while  the 
planes  of  one  pair  are  non- 
interchangeable  with  those 
of  the  other  pair  which  they 
intersect  at  an  angle  of  45°. 

Selection,  Position,  and  Des- 
ignation of  the  Crystal 
Axes. 

The  principal  symmetry 
axis  is  chosen  as  one  of  the 
crystal  axes,  is  held  verti- 
cally, and  is  called  the  c 

axis.      Two   other   crystal 

T         j    ' 
axes  are  so  selected  as  to 

coincide  with  one  set  of  interchangeable  secondary 
symmetry  axes.     One  is  held  horizontally  from  front 

86 


FIG.  56.  -  Crystal  axes  of 
the  tetragonal  system. 


HOLOHEDRAL  DIVISION  87 

to  back,  the  other  horizontally  from  right  to  left, 
and  both  are  called  a  axes,  since  they  are  interchange- 
able. Three  crystal  axes  intersecting  at  right  angles 
are,  then,  used  in  the  tetragonal  system,  as  in  the 
isometric,  and  they  are  held  in  the  same  positions  as 
in  the  isometric  system.  The  two  horizontal  axes 
are  interchangeable,  but,  unlike  the  conditions  in  the 
isometric  system,  neither  is  interchangeable  with  the 
vertical  axis  (see  Fig.  56). 

Orienting  Crystals. 

Holohedral  tetragonal  forms  are  oriented  by  hold- 
ing the  principal  symmetry  plane  horizontally,  and 
either  set  of  interchangeable  secondary  symmetry 
planes  vertically  from  front  to  back  and  from  right 
to  left.  The  crystal  axes  will  then  extend  in  the 
proper  directions. 

First  Order  Position  Defined. 

Forms  with  faces  or  faces  extended  that  cut  the 
two  horizontal  crystal  axes  equally  (at  equal  finite 
distances  from  the  origin)  are  said  to  be  in  the  first 
order  position. 

Second  Order  Position  Defined. 

Forms  with  faces  or  faces  extended  parallel  to  one 
(and  only  one)  horizontal  axis  are  said  to  be  in  the 
second  order  position. 

Third  Order  Position  Defined. 

Forms  with  faces  or  faces  extended  that  cut  the 
two  horizontal  axes  unequally  at  finite  distances  are 
said  to  be  in  the  third  order  position. 


88  TETRAGONAL  SYSTEM 

Holohedral  Tetragonal  Forms  Tabulated. 


Name. 

Symbol. 

Number  of 
faces. 

1st  order  pyramid  (Fig.  57)  

a  :  me 

8 

1st  order  prism  (Fig.  58)  
2nd  order  pyramid  (Fig.  59)  
2nd  order  prism  (Fig.  60)  

a  :  ooc 
ooo  :  me 
00  a  :  oo  c 

4 
8 
4 

Ditetragonal  pyramid  (Fig.  61)  

na  :  me 

16 

Ditetragonal  prism  (Fig  62) 

8 

Basal-pinacoid  (Fig.  63) 

2 

FIG.  57.  —  1st  order  pyramid.      FIG.  58.  —  1st  order  prism. 


FIG.  59.  —  2nd  order  pyramid.      FIG.  60.  —  2nd  order  prism. 


HOLOHEDRAL  DIVISION 


89 


FIG.  61.  —  Ditetragonal 
pyramid. 


FIG.  62.  —  Ditetragonal 
prism. 


FIG.  63.  —  Basal  pinacoid. 


Synonyms  for  the  Names 
of  the  Holohedral  Tet- 
ragonal Forms. 

1st    order    pyramid  - 

direct    pyramid,    1st 

order    bipyramid,   or 

unit  pyramid. 
1st  order  prism  —  direct 

prism  or  unit  prism. 
2nd    order    pyramid  — 

indirect    pyramid    or 

diametral  pyramid. 
2nd  order  prism  —  indirect  prism  or  diametral 

prism. 
Ditetragonal  pyramid  —  ditetragonal  bipyramid 

or  zirconoid. 

Ditetragonal  prism  —  none. 
Basal-pinacoid  —  basal-plane. 

Method  of  Determining  Holohedral  Tetragonal  Forms 
by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  select  any  face  in  the  upper  right 


90  TETRAGONAL  SYSTEM 

octant  facing  the  observer,  and  ascertain  the  relative 
distances  at  which  its  plane  intersects  the  three 
crystal  axes,  remembering  that  no  face  or  face 
extended  can  cut  the  vertical  axis  at  the  same 
distance  from  the  origin  as  it  cuts  either  horizontal 
axis.  If,  for  instance,  it  appears  that  the  plane  of 
the  face  selected  intersects  the  three  axes,  but  that 
the  two  horizontal  axes  are  not  cut  at  the  same 
distance  from  the  origin,  the  symbol  of  that  face 
(and  of  the  form  of  which  it  is  a  part)  is  a  :  na  :  me. 
By  referring  to  the  table  of  holohedral  tetragonal 
forms  (which  should  be  memorized  as  soon  as  possi- 
ble) it  is  seen  that  the  form  is  the  di tetragonal 
pyramid.  If  more  than  one  form  is  represented  on 
the  crystal,  each  may  be  determined  in  the  same  way. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

The  holohedral  tetragonal  forms  are  so  easily 
identified  after  they  have  been  properly  oriented 
that  it  seems  almost  unnecessary  to  offer  rules 
aiming  toward  their  rapid  recognition. 

The  following  statements  may,  however,  prove 
useful  to  the  beginner. 

1st  order  pyramid:  A  single  face  lying  wholly 
within  an  octant. 

1st  order  prism:  A  vertical  face  cutting  both  hori- 
zontal axes  equally. 

2nd  order  pyramid:  A  face  sloping  down  from  the 
vertical  axis  directly  towards  the  observer. 

The  2nd  order  pyramid  differs  in  no  way  from 
the  1st  order  pyramid  excepting  in  position  with 
respect  to  the  horizontal  crystal  axes;  and  an  8- 


HOLOHEDRAL  DIVISION  91 

faced  pyramid  may  be  placed  in  either  the  1st  or  2nd 
order  position  at  will.  Such  a  pyramid  may,  then, 
be  considered  either  a  1st  or  2nd  order  pyramid 
depending  upon  the  set  of  interchangeable  symmetry 
axes  with  which  the  crystal  axes  are  chosen  to  coin- 
cide. It  is  only  when  forms  in  both  the  1st  and  2nd 
order  positions  are  present  on  a  crystal  that  it  is 
necessary  to  distinguish  between  1st  and  2nd  order 
pyramids. 

2nd  order  prism:  A  vertical  face  extending  from 
right  to  left  or  front  to  back. 

As  is  the  case  with  the  2nd  order  pyramid,  a  2nd 
order  prism  differs  in  no  way  from  a  1st  order  prism 
excepting  in  position  with  respect  to  the  horizontal 
crystal  axes;  and  all  that  was  said  in  the  preceding 
section  relative  to  the  2nd  order  pyramid  applies 
with  equal  truth  to  the  2nd  order  prism. 

It  is  customary  to  select  the  horizontal  crystal 
axes  in  such  a  way  as  will  place  the  largest  and  most 
prominent  8-faced  pyramid  or  4-faced  prism  in  the 
1st  order  position. 

Pyramids  and  prisms  intersecting  in  horizontal 
edges  are  always  of  the  same  order. 

Ditetragonal  pyramid:  Two  identical  faces  lying 
wholly  within  an  octant. 

Ditetragonal  prism:  A  vertical  face  cutting  the  two 
horizontal  axes  unequally. 

Basal-pinacoid:  A  horizontal  face  on  top  of  the 
crystal. 

Fixed  and  Variable  Forms. 

The  only  fixed  holohedral  tetragonal  forms  are  the 
first  and  second  order  prism  and  the  basal-pinacoid. 


92  TETRAGONAL  SYSTEM 

Fixed  Angles  of  the  Tetragonal  System. 

The  only  fixed  angles  in  this  system  are  those 
between  the  fixed  forms  just  mentioned,  namely, 
90°  and  45°  (or  135°). 

Miscellaneous. 

The  general  statements  made  in  the  discussion  of 
the  holohedral  division  of  the  isometric  system 
regarding  combination  of  forms,  determination  of 
the  number  of  forms,  repetition  of  forms  on  a 
crystal,  and  limiting  forms  applies  with  equal  truth 
to  all  the  divisions  of  the  tetragonal  system.  It  may 
be  mentioned,  however,  that  repetitions  of  the  same 
variable  form  are  very  common  in  the  tetragonal 
system,  and  crystal  models  showing  such  repeated 
forms  are  not  difficult  to  obtain. 


A  B 

FIG.  64.  —  Holohedral  tetragonal  crystals. 
A:  Basal-pinacoid  (c),   1st  order  pyramid  (r),   1st  order 
prism  (m),  2nd  order  prism  (a),  and  ditetragonal  prism  (h). 

B:  Basal-pinacoid  (c),  two  1st  order  pyramids  (p  and  t\ 
1st  order  prism  (m),  2nd  order  prism  (a),  ditetragonal  pyramid 
(i),  and  ditetragonal  prism  (h). 


SPHENOIDAL  HEMIHEDRAL  DIVISION      93 

SPHENOIDAL  HEMIHEDRAL  DIVISION 
Development  or  Derivation  of  the  Forms. 

Sphenoidal  hemihedral  forms  may  be  conceived  to 
be  developed  by  dividing  each  of  the  holohedral 
forms  by  means  of  the  principal  symmetry  plane 
and  the  set  of  secondary  symmetry  planes  containing 
the  crystal  axes  into  eight  parts,  or  octants,  then 
suppressing  all  faces  lying  wholly  within  alternate 
!  parts  thus  obtained,  and  extending  the  remaining 
faces  until  they  meet  in  edges  or  corners. 

Symmetry. 

Sphenoidal  hemihedral  forms  possess  only  two 
interchangeable  secondary  symmetry  planes  at  right 
angles  to  each  other. 

Selection,   Position,   and  Designation  of  the  Crystal 
Axes. 

The  three  directions  used  as  crystal  axes  in  the 
holohedral  division  are  still  utilized  for  the  same 
purpose  in  the  sphenoidal  hemihedral  division.  In 
other  words,  the  vertical  or  c  axis  lies  at  the  inter- 
section of  the  two  secondary  symmetry  planes; 
while  the  two  interchangeable  horizontal  or  a  axes, 
one  of  which  extends  from  front  to  back,  and  the 
other  from  right  to  left,  make  an  angle  of  90°  with 
each  other,  and  bisect  the  angles  between  the  two 
secondary  symmetry  planes. 

Orienting  Crystals. 

Sphenoidal  hemihedral  crystals  are  oriented  in 
exactly  the  same  way  as  are  tetrahedral  hemihedral 
isometric  ones  (see  p.  30). 


94  TETRAGONAL  SYSTEM 

Sphenoidal  Hemihedral  Tetragonal  Forms  Tabulated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

rtTetragonal  sphenoid  (Fig.  / 
65)  j 
=t  Tetragonal     scalenohedronl 

a:  a:  me 

*nr 

o  :  no  :  me 

4 

1st  order  pyramid 

(Fig.  66)  ) 

i       2 

1st  order  prism  (Fig.  58)  

a  :  a  :  ooc 
2 
o  :  oo  o  :  me 

4 

1st  order  prism 

2 
a:  ooa:  <»c 

2 

a  :  na  :  oo  c 

2 

ooa  :  ooo  :  c 

2 

FIG.  65.  —  Positive  (on  left)  and  negative  (on  right)  tet- 
ragonal sphenoids  containing  the  forms  from  which  they  are 
derived.  The  suppressed  faces  are  shaded. 

Synonyms  for  the  Names  of  the  Sphenoidal  Hemi- 
hedral Tetragonal  Forms. 

Tetragonal  sphenoid  —  hemi-unit  pyramid. 
Tetragonal  scalenohedron  —  none. 


SPHENOIDAL  HEMIHEDRAL  DIVISION      95 

Positive  and  Negative  Forms  in  the  Sphenoidal  Hemi- 
hedral  Tetragonal  Division. 

+  and  —  forms  are  recognized  in  this  division, 
and  are  distinguished  in  exactly  the  same  way  as  are 
the  -f  and  —  forms  in  the  tetrahedral  hemihedral 


FIG.  66.  —  Positive  (on  left)  and  negative  (on  right)  tet- 
ragonal scalenohedrons  containing  the  forms  from  which  they 
are  derived.  The  suppressed  faces  are  shaded. 

division  of  the  isometric  system  (see  p.  32).  All 
that  was  said  there  relative  to  such  forms  will  apply 
with  equal  truth  to  the  +  and  —  forms  in  the 
division  under  consideration. 

Methods     of    Determining     Sphenoidal    Hemihedral 
Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  all  the  forms  but  the  tetragonal 
sphenoid  and  scalenohedron  may  be  identified  easily 
by  applying  the  rules  already  given  for  the  deter- 
mination of  holohedral  forms  of  the  same  name. 

The  tetragonal  sphenoid  and  scalenohedron  may 
be  recognized  by  determining  the  symbol  of  any  face 
in  the  manner  described  in  the  discussion  of  holo- 
hedral forms,  dividing  this  symbol  by  2,  and  then 
ascertaining  from  the  table  the  name  of  the  form 
possessing  this  symbol. 


96  TETRAGONAL  SYSTEM 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  Call  the  forms  +  or  —  according  to  the 
rules  already  set  forth.  It  is  assumed  that  the 
student  is  already  familiar  with  the  rules  for  recog- 
nizing those  forms  identical  in  shape  with  the  holo- 
hedral  ones  (see  p.  90). 

Tetragonal  sphenoid:  A  single  face  in  an  octant 
(although  not  necessarily  wholly  included  therein) 
sloping  down  from  the  vertical  axis  in  such  a  way  as 
to  cut  both  horizontal  axes  equally. 

Tetragonal  scalenohedron:  Two  faces  lying  within 
an  octant  (although  not  necessarily  wholly  included 
therein)  which  cut  all  three  axes  unequally. 


B 

FIG.  67.  —  Sphenoidal  hemihedral  tetragonal  crystals. 
A:    -f-  and  —  tetragonal  sphenoids  (p  and  pi)  and  2nd  order 
pyramid  (z). 

B:  Basal  pinacoid  (c),  two  2nd  order  pyramids  (e  and  z), 
1st  order  prism  (ra),  +  tetragonal  sphenoid  (p),  and  tetrag- 
onal scalenohedron  (s). 


PYRAMIDAL  HEMIHEDRAL  DIVISION       97 

PYRAMIDAL  HEMIHEDRAL   DIVISION 

Development  or  Derivation  of  the  Forms. 

Pyramidal  hemihedral  forms  may  be  conceived  to 
be  developed  by  dividing  each  of  the  holohedral 
forms  by  means  of  all  four  secondary  symmetry 
planes  into  eight  parts,  then  suppressing  all  faces 
lying  wholly  within  alternate  parts  thus  obtained, 
and  extending  the  remaining  faces  until  they  meet 
in  edges  or  corners. 

Symmetry. 

Pyramidal  hemihedral  forms  possess  only  one 
symmetry  plane  which  is  in  the  position  of  the 
principal  symmetry  plane  existing  in  the  holohedral 
division.  It  is,  however,  in  the  pyramidal  hemi- 
hedral division  a  secondary  rather  than  a  principal 
symmetry  plane  since  there  are  no  interchangeable 
symmetry  planes  perpendicular  to  it., 

Pyramidal  hemihedral  tetragonal  forms  may, 
then,  be  said  to  be  characterized  by  the  presence  of 
one  secondary  symmetry  plane  and  a  general  four- 
fold arrangement  of  the  faces. 

Selection,  Position,  and  Designation  of  the   Crystal 
Axes. 

The  vertical  or  c  axis  is  made  to  coincide  with  the 
secondary  symmetry  axis.  Two  interchangeable  hor- 
izontal axes  parallel  to  prominent  crystallographic 
directions  at  right  angles  to  each  other  are  also 
selected,  one  of  which  is  so  placed  as  to  extend  from 
front  to  back,  and  the  other  from  right  to  left. 
Being  interchangeable,  both  are  called  a  axes. 


98 


TETRAGONAL  SYSTEM 


Orienting  Crystals. 

The  secondary  symmetry  plane  is  held  horizon- 
tally. The  crystal  is  then  rotated  around  the 
symmetry  axis  until  the  most  prominent  pyramid 
or  prism  lies  in  the  first  order  position.  The  crystal 
axes  will  then  extend  in  the  proper  directions. 

Pyramidal  Hemihedral  Tetragonal  Forms  Tabulated. 


Name. 

Symbol. 

Num- 
ber of 
faces. 

Form  from  which 
derived. 

3rd  order  pyramid  (Fig.  68)  ... 
3rd  order  prism  (Fig.  69)  

1st  order  pyramid  (Fig.  57)  ... 
1st  order  prism  (Fig.  68)  
2nd  order  pyramid  (Fig.  59)  .  . 

4^ 

2nd  order  prism  (Fig.  60)  
Basal-pinacoid  (Fig.  63)  

a  :  na  :  me 

8 
4 

8 
4 
8 
4 
2 

ditetragonal  pyramid 
ditetragonal  prism 

1st  order  pyramid 
1st  order  prism 
2nd  order  pyramid 
2nd  order  prism 
basal-pinacoid 

2 

o  :  no  :  oo  c 

2 

a  :  a  :  me 

2 

a  :  a:  ooc 

2 
o  :  oo  a  :  me 

2 

a  :  oo  a  :  oo  c 

2 

ooa  :  ooa  :  c 

2 

FIG.  68.  —  3rd  order  pyra-  FIG.  69.  —  3rd  order  prism 
mid  containing  the  form  from  containing  the  form  from 
which  it  is  derived.  Sup-  which  it  is  derived.  Sup- 
pressed faces  are  shaded.  pressed  faces  are  shaded. 


PYRAMIDAL  HEMIHEDRAL  DIVISION       99 

Synonyms  for  the  Names  of  the  Pyramidal  Hemihedral 
Tetragonal  Forms. 

3rd  order  pyramid  —  square  pyramid  of  the  third 
order  or  third  order  bipyramid. 

3rd  order  prism  —  square  prism  of  the  third  order. 

Method  of  Determining  Pyramidal  Hemihedral  Tetrag- 
onal Forms  by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner 
already  described  all  the  forms  but  the  3rd  order 
pyramid  or  prism  may  be  identified  easily  by 
applying  the  rules  already  given  for  the  determina- 
tion of  holohedral  forms  of  the  same  name.  The 
3rd  order  pyramid  or  prism  may  be  recognized  by 
determining  the  symbol  of  any  face  in  the  manner 
already  described  in  the  discussion  of  holohedral 
isometric  forms,  dividing  this  symbol  by  2,  and  then 
ascertaining  from  the  table  the  name  of  the  form 
possessing  this  symbol. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen.  It  is  possible,  but  unnecessary,  to  dis- 
tinguish between  positive  and  negative  forms  in  this 
division.  It  is  assumed  that  the  student  is  already 
familiar  with  the  rules  for  recognizing  those  forms 
identical  in  shape  and  position  with  the  holohedral 
ones  (see  p.  90). 

3rd  order  pyramid:  A  single  face  lying  in  an  octant 
(although  not  necessarily  wholly  included  therein) 


100 


TETRAGONAL  SYSTEM 


which  intersects  all  three  crystal  axes  at  different 
finite  distances  from  the  origin. 

The  3rd  order  pyramid  differs  in  no  way  from  the 
1st  or  2nd  order  pyramid  excepting  in  position  with 
respect  to  the  horizontal  crystal  axes.  All  three 
types  of  8-faced  pyramids  may  have  the  same 
appearance;  and  any  such  pyramid  may  be  held  at 
will  as  a  1st,  2nd,  or  3rd  order  pyramid.  The  3rd 
order  pyramid  is  skewed  or  twisted  through  a  small 


FIG.  70.  —  Diagram 
showing  the  relations 
of  the  1st  order  (dotted 
lines),  2nd  order 
(broken  lines),  and  3rd 
order  (solid  lines)  pyra- 
mids and  prisms  to  the 
horizontal  crystal  axes. 


A  B 

FIG.  71.  —  Pyramidal  hemihedral 

hexagonal  crystals. 
A:   1st  order  pyramid  (r),  1st  or- 
der prism  (ra),  2nd  order  prism  (a), 
and  3rd  order  pyramid  (z). 

B:  Basal-pinacoid  (c),  1st  order 
pyramid  (ft),  1st  order  prism  (m), 
and  3rd  order  prism  (/). 


angle  (less  than  45°)  either  to  the  right  or  left  away 
from  the  position  of  the  1st  or  2nd  order  pyramid. 
Fig.  70  shows  how  the  horizontal  axes  are  cut  by 
1st,  2nd,  and  3rd  order  pyramids  and  prisms. 

3rd  order  prism:  A  single  face  parallel  to  the 
vertical  axis,  which  intersects  the  two  horizontal 
axes  at  unequal  distances  from  the  origin. 

All  that  was  said  in  the  preceding  section  relative 
to  the  3rd  order  pyramid  applies  with  equal  truth 
to  the  3rd  order  prism. 


PYRAMIDAL  HEMIHEDRAL  DIVISION     101 


Application  of  the  Law  Go7erning  Combination  of 
Forms. 

It  has  already  been  noted  (see  p.  41)  that  one 
of  the  commonest  mistakes  made  in  determining 
crystal  forms  is  the  naming  of  two  or  more  forms 
which  cannot  possibly  occur  on  the  same  crystal, 
such,  for  instance,  as  the  sphenoid  and  3rd  order 
pyramid.  The  presence  on  the  same  crystal  of  two 
forms  like  these,  which  are  peculiar  to  different 
divisions  of  the  system,  is,  of  course,  in  direct  vio- 
lation of  the  law  governing  the  combination  of 
forms  (see  p.  36).  A  student  should  thoroughly 
familiarize  himself  with  the  following  table  if  he 
wishes  to  avoid  the  mistake  mentioned. 


Holohedral  forms. 

Corresponding  sphenoi- 
dal  hemihedral  forms. 

Corresponding  pyra- 
midal hemihedral 
forms. 

1st  order  pyramid  
1st  order  prism  

tetragonal  sphenoid 
1st  order  prism 

1st  order  pyramid 
1st  order  prism 

2nd  order  pyramid  
2nd  order  prism       .   ... 

2nd  order  pyramid 
2nd  order  prism 

2nd  order  pyramid 
2nd  order  prism 

Ditetragonal  pyramid.  .  .  . 
Ditetragonal  prism  

tetragonal  scalenohedron 
ditetragonal  prism 

3rd  order  pyramid 
Srd  order  prism 

Basal-pinacoid  

basal-pinacoid 

basal-pinacoid 

It  will  be  noted  from  the  above  table  that  the 
1st  order  prism,  2nd  order  pyramid  and  prism,  and 
the  basal  pinacoid  occur  in  all  three  of  the  divisions 
already  discussed,  and  may,  therefore,  be  combined 
with  any  other  forms  in  these  divisions.  Further, 
it  will  be  seen  that  the  1st  order  pyramid  and 
ditetragonal  prism  occur  unchanged  in  name  or 
shape  in  two  of  the  divisions ;  while  the  ditetragonal 
pyramid  occurs  only  as  a  holohedral  form. 


102 


1  TETRAGONAL  SYSTEM 


TRAPEZOHEDRAL   HEMIHEDRAL   DIVISION 

Trapezohedral  hemihedral  forms  may  be  conceived  to  be 
developed  by  dividing  each  holohedral  form  by  the  principal 
and  all  the  secondary  symmetry  planes  into  sixteen  parts, 
then  suppressing  all  faces  lying  wholly  within  alternate  parts 
thus  obtained,  and  extending  the 
remaining  faces  until  they  meet  in 
edges  or  corners. 

As  the  ditetragonal  pyramid  is 
the  only  tetragonal  form  with  six- 
teen faces,  it  is  evident  that  a  dite- 
tragonal pyramid  face  is  the  only 
one  that  can  lie  wholly  within  one 
of  the  parts  obtained  by  dividing  a 
tetragonal  crystal  in  the  manner 
just  specified.  The  ditetragona  ing  the  form  from 
pyramid  is,  then,  the  only  tetragonal  -t  ig  deriyed  The 

form  from  which  a  trapezohedral  pressed  faces  are  shaded, 
hemihedral  form  differing  from  the 

holohedral  one  in  shape  and  in  name  can  be  derived.  This 
new  form  is  called  the  tetragonal  trapezohedron  (Fig.  72). 
Since  no  mineral  is  known  to  crystallize  in  this  division,  its 
further  consideration  seems  unnecessary. 


FIG.  72. 


Tetragonal 
contain- 


Table   of   Holohedral   Tetragonal    Symbols   Used   by 
Various  Authorities. 


Weiss. 

Naumann. 

Dana. 

Miller. 

1st  order  pyramid  
1st  order  prism  

:  a:  me 

'.  a  '  oo  c 

mP 

OOP 

m 
I 

Ill 

110 

2nd  order  pyramid     .   . 

:  oo  o  :  me 

mP  oo 

m-i 

hOl 

2nd  order  prism 

•  oo  a  :  oo  c 

ooPoo 

100 

Ditetragonal  pyramid  
Ditetragonal  prism 

:  na  :  me 
'  na  '  oo  c 

mPn 
oo  Pn 

m-n 

hkl 

hkO 

Basal-pinacoid 

oo  a  •  oo  a  :  c 

OP 

o 

001 

For  methods  of  forming  hemihedral  symbols,  see  page  44. 


CHAPTER  V 
OBTHOBHOMBIC    SYSTEM 

HOLOHEDRAL   DIVISION 
Symmetry. 

The  holohedral  division  of  the  orthorhombic 
system  is  characterized  by  the  presence  of  three 
non-interchangeable  secondary  symmetry  planes  at 
right  angles  to  each  other. 

The  Selection,  Position,  and  Designation  of  the  Crystal 
Axes. 

The  three  crystal  axes  utilized  in  this  system  are  so 
chosen  as  to  coincide  with  the  secondary  symmetry 


FIG.  73.  —  Crystal  axes  of  the  orthorhombic  system. 

axes.     One  is  held  vertically  and  is  called  the  vertical 

or  c  axis;  another  is  held  horizontally  from  right  to 

left,  and  is  called  the  macro  (long)  or  6  axis;  while 

103 


104  ORTHORHOMBIC  SYSTEM 

the  third  extends  horizontally  from  front  to  back, 
and  is  called  the  brachy  (short)  or  a  axis.  None 
of  the  three  crystal  axes  are  interchangeable  (see 
Fig.  73). 

Orienting  Crystals. 

One  symmetry  plane  is  held  so  as  to  extend  ver- 
tically from  front  to  back,  another  vertically  from 
right  to  left,  and  the  third  horizontally.  The  crys- 
tal axes  will  then  extend  in  the  proper  directions, 
and  the  forms  can  be  named  according  to  the 
directions  given  later.  There  are,  however,  certain 
conventions  that  are  set  forth  in  immediately  suc- 
ceeding paragraphs,  and  which  should  be  observed 
as  closely  as  possible.  In  considering  the  statements 
that  follow,  it  should  be  remembered  that  the  rela- 
tive lengths  of  the  axes  are  determined  by  the  dis- 
tances from  the  origin  to  the  points  where  the 
plane  of  a  face  of  the  ground-form  intersects  each 
crystal  axis. 

If  a  crystal  is  decidedly  elongated,  the  longest 
axis  becomes  the  c  axis;  while,  if  it  is  notably  tabu- 
lar, the  shortest  axis  is  used  as  the  c  axis.  When 
neither  elongated  nor  tabular  an  axis  of  intermedi- 
ate length  is  used  for  the  c  axis. 

The  c  axis  having  been  selected,  the  longer  of  the 
other  two  axes  is  held  from  right  to  left  as  the 
macro  or  b  axis,  and  the  shorter  from  front  to  back 
as  the  brachy  or  a  axis. 

Since  the  student  is  unable  to  determine  which 
form  is  the  ground-form,  and  since  the  ground-form 
may  not  be  represented  on  some  crystals,  it  is  per- 
missible to  determine  the  relative  lengths  of  the 


HOWHEDRAL  DIVISION 


105 


axes  by  noting  the  distances  from  the  origin  at 
which  prominent  faces  intersect  two  or  three  axes. 

Holohedral  Orthorhombic  Forms  Tabulated. 


Name. 

Symbol. 

Number  of 
faces. 

Pyramid  (Fig   74) 

na  '  b  '.  me 

8 

Prism  (Fig.  75) 

4 

Macro-dome  (Fig   76) 

4 

Brachy-dome  (Fig   77) 

4 

Macro-pinacoid  (Fig   78) 

a  '  oo  6  •  oo  c 

2 

Brachy-pinacoid  (Fig.  79)  

QOO:  6  :  ooc 

2 

Basal-pinacoid  (Fig.  80)  

oo  a  :  oo  6  :  c 

2 

FIG.  74.  —  Pyramid. 


FIG.  75.  —  Prism. 


FIG.  76.  —  Macro-dome. 


FIG.  77.  —  Brachy-dome. 


106 


ORTHORHOMBIC  SYSTEM 


Remarks  on  the  Holohedral  Orthorhombic  Forms. 

It  will  be  seen  by  examining  the  table  just  given 
that  the  holohedral  orthorhombic  forms  may  be 
grouped  into  three  divisions,  one  containing  the 
pyramid  with  eight  faces  cutting  all  three  axes  at 


FIG.  78.  —  Macro-pinacoid.         FIG.  79.  —  Brachy-pinacoid. 


-4-'- 


FIG.  80.  —  Basal-pinacoid. 


finite  distances,  one  con- 
taining the  prism  and 
domes  with  four  faces 
parallel  to  one  axis  and 
cutting  the  other  two  at 
finite  distances,  and  one 
containing  the  pinacoids 
with  two  faces  parallel  to 
two  axes.  The  forms  in  any  one  of  these  divisions 
may  be  changed  into  any  other  form  in  the  same 
division  by  a  different  selection  of  crystal  axes;  but 
a  form  in  no  one  division  can  be  named  as  a  form 
in  any  of  the  other  divisions  no  matter  how  the 
crystal  is  held.  The  pyramid  is  the  only  holohedral 
orthorhombic  form  which  will  when  unmodified 
completely  bound  the  crystal.  Since  all  the  forms 
but  the  pinacoids  (the  fixed  forms)  contain  one  or 
more  variables  in  their  symbols,  they  may  be  re- 
peated an  indefinite  number  of  times  on  the  same 
crystal. 


HOLOHEDRAL  DIVISION  107 

Domes  and  pinacoids  (excepting  the  basal-pina- 
'coid)  are  named  by  prefixing  the  name  of  the  hori- 
zontal axis  to  which  they  are  parallel. 

Synonyms  for  the  Names  of  the  Holohedral  Ortho- 
rhombic  Forms. 

Pyramid  —  unit-pyramid,    macro-pyramid,    and 

brachy-pyramid. 
Prism  —  unit-prism,    macro-prism,    and    brachy- 

prism. 

Macro-dome  —  none. 
Brachy-dome  —  none. 
Macro-pinacoid  —  none. 
Brachy-pinacoid  —  none. 
Basal-pinacoid  —  basal-plane. 

Method    of    Determining    Holohedral    Orthorhombic 
Forms  by  the  Use  of  Symbols. 

The  usual  method  of  determining  forms  by  the 
use  of  symbols,  as  presented  in  the  discussion  of 
the  systems  already  described,  may  be  used  success- 
fully in  the  orthorhombic  system.  In  forming 
symbols  it  should  be  remembered  that  the  distance 
from  the  origin  to  the  point  where  the  plane  of  any 
face  intersects  the  macro  or  b  axis  is  always  made  b 
(or  006  if  the  face  is  parallel  to  this  axis);  while 
the  distance  from  the  origin  to  the  point  where  the 
plane  of  the  face  intersects  the  brachy  or  a  axis  is 
called  na  (or  ooa)  if  the  plane  cuts  the  b  axis  at  a 
finite  distance  from  the  origin,  and  is  called  a  (or  oo  a) 
if  the  plane  cuts  the  b  axis  at  infinity.  Similarly, 
the  distance  from  the  origin  to  the  point  where  the 
plane  of  a  face  intersects  the  c  axis  is  called  me 
(or  oo  c)  if  the  plane  cuts  either  the  a  or  b  axes  at 


108  ORTHORHOMBIC  SYSTEM 

finite  distances;    while  it  is  called  c  if  both  the  a 
and  b  axes  are  cut  at  infinity. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  (which  should  be  learned  at 
once)  apply  to  the  face  or  faces  of  different  shape  or 
size  seen. 

Pyramid:  A  face  lying  entirely  within  an  octant. 

Prism:  A  vertical  face  oblique  to  both  horizontal 
axes. 

Macro-dome:  A  face  sloping  from  the  vertical 
axis  directly  down  toward  the  observer. 

Brachy-dome:  A  face  sloping  from  the  vertical 
axis  down  to  the  right  or  left. 

Macro-pinacoid:  A  vertical  face  extending  from 
right  to  left. 

Brachy-pinacoid:  A  vertical  face  extending  from 
front  to  back. 

Basal-pinacoid:  A  horizontal  face  on  top  of  the 
crystal. 

Fixed  and  Variable  Forms. 

The  only  fixed  holohedral  orthorhombic  forms  are 
the  three  pinacoids. 

Fixed  Angles  of  the  Orthorhombic  System. 

The  only  fixed  angle  in  this  system  is  that  between 
the  three  pinacoids,  namely,  90°. 

Miscellaneous. 

The  general  statements  made  in  the  discussion 
of  the  holohedral  division  of  the  isometric  system 


SPHENOIDAL  HEMIHEDRAL  DIVISION     109 

regarding  combination  of  forms,  determination  of 
the  number  of  forms,  and  limiting  forms  apply 
with  equal  truth  to  all  the  divisions  of  the  tetragonal 
system.  Hemimorphism  (see  p.  17)  is  shown  by 
crystals  of  calamine  as  well  as  by  those  of  certain 
rare  minerals. 


A  B 

FIG.  81.  —  Holohedral  orthorhombic  crystals. 
A:  Basal-pinacoid   (c),   macro-dome    (d),   macro-pinacoid 
(a),  brachy-dome  (o),  brachy-pinacoid  (6),  pyramid  (y),  and 
two  prisms  (ra  and  ri). 

B  (hemimorphic) :  Basal-pinacoid  (c),  brachy-dome  (e), 
brachy-pinacoid  (6),  macro-dome  (0,  and  prism  (m).  On 
other  end:  pyramid  (v). 


SPHENOIDAL  HEMIHEDRAL  DIVISION 

Development  or  Derivation  of  the  Forms. 

Sphenoidal  hemihedral  forms  may  be  conceived  to  be  de- 
veloped by  dividing  each  of  the  holohedral  forms  by  means  of 
the  three  secondary  symmetry  planes  into  eight  parts,  then 
suppressing  all  faces  lying  wholly  within  alternate  parts  thus 
obtained,  and  extending  the  remaining  faces  until  they  meet 
in  edges  or  corners. 

Symmetry. 

Sphenoidal  hemihedral  forms  possess  no  symmetry  planes 
whatever,  but  are  characterized  by  the  presence  of  three 


110 


ORTHORHOMBIC  SYSTEM 


prominent,  non-interchangeable  crystallographic  directions  at 
right  angles  to  each  other. 

Selection,   Position,   and  Designation  of  the  Crystal 
Axes. 

Three  prominent,  non-interchangeable  crystallographic 
directions  at  right  angles  to  each  other  are  selected  as  the 
crystal  axes.  These  are  held  and  named  exactly  as  in  the 
holohedral  division. 

Orienting  Crystals. 

Usually  the  easiest  way  to  orient  sphenoidal  hemihedral 
crystals  is  to  identify  by  general  appearance  some  form  whose 
holohedral  and  hemihedral  shapes  are  the  same,  and  to  hold 
this  form  in  the  position  it  occupies  in  the  holohedral  division. 
It  is  often  just  as  easy  or  easier  to  find  three  prominent,  non- 
interchangeable  crystallographic  directions  at  right  angles 
to  each  other,  and  to  hold  these  in  the  positions  of  the  crystal 
axes,  in  the  manner  already  set  forth  in  the  discussion  of  the 
holohedral  division. 

Sphenoidal  Hemihedral  Orthorhombic  Forms  Tabu- 
lated. 


Name. 

Symbol.1 

Number 
of  faces. 

Form  from 
which  derived. 

iOrthorhombic  sphenoid  (Fig.( 

no  :  6  :  me 

82)               ) 

-1-        2 

na  :  b  :  oo  c 

4 

Macro-dome  (Fig.  76)  

2 
a:oob:  me 

4 

macro-do  me 

2 
oo  a  :  6  :  me 

4* 

2 
a  :  oo  6  :  oo  c 

2 

macropinacoid 

2 
ooa  :  6  :  ooc 

2 

2 
ooa  :  006:  c 

2 

SPHENOID AL  HEMIHEDRAL  DIVISION      111 

Synonyms  for  the  Names  of  Sphenoidal  Hemihedral 
Tetragonal  Forms. 

Orthorhombic  sphenoid  —  none. 

Positive  and  Negative  Forms  in  the  Sphenoidal  Hemi- 
hedral Division. 

+  and  —  forms  are  recognized  in  this  division,  and  are 
distinguished  in  exactly  the  same  way  as  are  the  +  and  — 
forms  in  the  tetrahedral  hemihedral  division  of  the  isometric 
system  (see  p.  32).  All  that  was  said  there  relative  to  such 
forms  will  apply  with  equal  truth  to  the  +  and  —  forms  in 
the  division  under  consideration. 


FIG.  82.  —  Positive  (on  left)  and  negative  (on  right)  ortho- 
rhombic  sphenoids  containing  the  forms  from  which  they  are 
derived.  The  suppressed  faces  are  shaded. 

Method  of  Determining  Sphenoidal  Hemihedral  Forms 
by  the  Use  of  Symbols. 

After  properly  orienting  the  crystal  in  the  manner  already 
described  all  the  forms  but  the  orthorhombic  sphenoid  may 
be  identified  easily  by  applying  the  rules  already  given  for 
the  determination  of  holohedral  forms  of  the  same  name. 

The  orthorhombic  sphenoid  may  be  recognized  by  deter- 
mining the  symbol  of  any  face  in  the  manner  described  in  the 
discussion  of  holohedral  forms,  dividing  this  symbol  by  two, 
and  then  ascertaining  from  the  table  the  name  of  the  form 
possessing  this  symbol. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

After  properly  orienting  the  crystal  all  the  forms  but  the 
orthorhombic  sphenoid  may  be  recognized  by  the  rules  al- 


112 


ORTHORHOMBIC  SYSTEM 


ready  given  for  identifying  the  holohedral  forms  (see  p.  108). 
The  following  description  of  the  position  of  an  orthorhombic 
sphenoid  face  should  be  learned  at  once;  and  if  faces  an- 
swering to  this  description  are  present  on  a  crystal,  they 
should  be  called  +  or  —  according  to  the  rules  already 
set  forth. 

Orthorhombic  Sphenoid:  A  single  face  in  an  octant  (al- 
though not  necessarily  wholly  included  therein)  sloping  down 
from  the  vertical  axis  so  as  to  cut  both  horizontal  axes 
obliquely. 

Table  of  Holohedral  Orthorhombic  Symbols  Used  by 
Various  Authorities. 


Weiss. 

Naumann. 

Dana. 

Miller. 

Pyramid 

na  '  b  '  me 

m  or  mPn 

1  or  m-n 

Ill  or  hkl 

na  '  b  '  oo  c 

mP  or  oo  Pn 

I  or  i-n 

110  or  hkO 

a  •  oo  6  :  me 

mP  oo 

m-i 

hOl 

Brachy-dome 

oo  a  •  b  :  me 

mP  So 

m-i 

Gkl 

Macropinacoid  

a  :  006:  °oc 

ooPw" 

t-t 

100 

Brachy-pinacoid  
Basal-pi  nacoid  

ooa  :6  :  we 
ooa  :  006  :  c 

ooPSo 
OP 

t-T 
O 

010 
001 

For  methods  of  forming  hemihedral  symbols,  see  p.  44. 


CHAPTER  VI 
MONOCLJNIC  SYSTEM 

HOLOHEDRAL   DIVISION 
Symmetry. 

The  holohedral  division  of  the  monoclinic  system 
is  characterized  by  the  presence  of  one  secondary 
symmetry  plane. 

Selection,  Position,  and  Desig- 
nation of  the  Crystal  Axes. 

The  ortho  or  6  crystal  axis 
is  made  to  coincide  with  the 
secondary  symmetry  axis, 
and  is  held  horizontally  from 
right  to  left.  The  other  two 
axes  (which  are  made  to  pass 
through  the  geometric  center  ^  g3  J  Crygtal  axeg 
of  the  crystal)  are  selected  of  the  monociinic  system, 
lying  in  the  symmetry  plane  p  is  a  variable  angle,  but 
parallel  to  two  prominent  can  never  be  equal  to  a 
crystallographic  directions  as  fi>e^  angle  of  any  system- 
nearly  at  right  angles  to  each  other  as  possible. 
One  is  held  vertically  and  called  the  vertical  or  c 
axis;  while  the  other  is  held  so  as  to  slope  or  in- 
cline down  toward  the  observer,  and  is  called  the 
clino  or  a  axis  (see  Fig.  83). 

None  of  the  crystal  axes  are  interchangeable. 

The  ortho  axis  makes  right  angles  with  both  the 
113 


114 


MONOCLINIC  SYSTEM 


vertical  and  the  clino  axes,  but  the  clino  and  vertical 
axes  are  never  exactly  at  right  angles  to  each  other. 

Orienting  Crystals. 

The  symmetry  plane  is  held  so  as  to  extend  verti- 
cally from  front  to  back.  The  crystal  is  then  rotated 
around  the  symmetry  (ortho)  axis  until  the  most 
prominent  crystallographic  direction  (which  fixes  the 
position  of  the  c  axis)  is  held  vertically,  and  a  second 
prominent  crystallographic  direction  (which  fixes  the 
position  of  the  clino  axis)  is  held  so  as  to  slope  or 
incline  down  toward  the  observer. 

Prominent  crystallographic  directions  may  be  edges, 
the  intersection  of  the  symmetry  plane  and  faces, 
or  lines  connecting  opposite  corners  or  the  middle 
points  of  opposite  edges  or  faces.  It  will  usually  be 
found  desirable  to  select  the  vertical  and  clino  axes 
parallel  to  prominent  edges,  unless  by  so  doing  the 
two  axes  mentioned  are  forced  to  intersect  in  a 
decidedly  acute  angle. 

Holohedral  Monoclinic  Forms  Tabulated. 


Name. 

Symbol. 

Number  of 
faces. 

+Pyramid  (Fig.  84)  

—  no  :  b  '.  me 

4 

—Pyramid  (Fig.  85)  

-\-na  '.  6  :  me 

4 

Prism  (Fig.  86)  

no  :  6  :  oo  c 

4 

Clino-dome  (Fig.  87)  

oo  a  •  b  :  me 

4 

Ortho-pinacoid  (Fig.  88)  

2 

+Ortho-dome  (Fig.  89)  

2 

—Ortho-dome  (Fig.  90)  

2 

Basal-pinacoid  (Fig.  91  )  

2 

Clino-pinacoid  (Fig.  92)  

oo  a  •  b  '  oo  c 

2 

HOLOHEDRAL  DIVISION 


115 


FIG.  84.  —  Positive  pyramid.      FIG.  85.  —  Negative  pyramid. 


FIG.  86. 
Prism. 


FIG.  87. 
Clino-dome. 


FIG.  88. 
Ortho-pinacoid. 


FIG.  89.  —  Positive 
ortho-dome. 

/UWAAA/WVVV) 


FIG.  90.  —  Negative 
ortho-dome. 


Fro.  91.  — Basal-pinacoid.         FIG.  92.  —  Clino-pinacoid. 


116  MONOCLINIC  SYSTEM 

Remarks  on  the  Holohedral  Monoclinic  Forms. 

It  will  be  seen  by  examining  the  table  just  given 
that  the  holohedral  monoclinic  forms  may  be 
grouped  into  three  divisions,  one  containing  forms 
whose  faces  intersect  the  symmetry  plane  obliquely, 
namely,  the  +  and  —  pyramid,  clino-dome,  and 
prism;  another  containing  forms  whose  faces  are 
perpendicular  to  the  symmetry  plane,  namely,  the 
ortho-pinacoid,  +  and  —  ortho-dome,  and  basal- 
pinacoid;  and  a  third  containing  a  form  whose  faces 
are  parallel  to  the  symmetry  plane,  namely,  the 
clino-pinacoid.  The  forms  in  any  one  of  these 
divisions  may  be  changed  into  any  other  form  in  the 
same  division  by  a  different  selection  of  crystal 
axes;  but  a  form  in  no  one  division  can  be  named  as 
a  form  in  any  of  the  other  divisions  no  matter  how 
the  crystal  is  held. 

Since  all  the  forms  but  the  pinacoids  (the  fixed 
forms)  contain  one  or  more  variables  in  their  sym- 
bols, they  may  be  repeated  an  indefinite  number  of 
times  on  the  same  crystal.  No  one  form  in  this 
system  will  completely  bound  a  crystal.  In  other 
words,  at  least  two  forms  must  always  be  repre- 
sented on  a  monoclinic  crystal. 

Domes  and  pinacoids  (excepting  the  basal -pina- 
coid)  are  named  by  prefixing  the  name  of  the  hori- 
zontal axis  to  which  they  are  parallel. 


HOLOHEDRAL  DIVISION  117 

Synonyms  for  the  Names  of  the  Holohedral  Mono- 
clinic  Forms. 

Pyramid  —  unit-,  ortho-,  and  clino-hemipyramid. 

Prism  —  unit-,  ortho-,  and  clino-prism. 

Clino-dome  —  none. 

Ortho-pinacoid  —  none. 

Ortho-dome  —  none. 

Basal-pinacoid  —  basal-plane. 

Clino-pinacoid  —  none. 

Positive  and  Negative  Forms  in  the  Holohedral  Mono- 
clinic  Division. 

The  +  and  —  forms  in  this  system  are  distin- 
guished in  quite  a  different  manner  from  the  +  and 
—  forms  in  any  of  the  systems  already  discussed, 
and  are  apt  to  prove  quite  confusing  to  a  student 
until  he  is  thoroughly  familiar  with  the  conceptions 
upon  which  their  distinction  is  based. 

It  has  been  decided  to  call  those  pyramids  and 
ortho-domes  —  whose  faces  or  faces  extended  inter- 
sect the  clino  axis  in  front  of  the  origin  (at  -\-na) ; 
while  those  pyramids  and  ortho-domes  whose  faces 
or  faces  extended  intersect  the  clino  axis  behind  the 
origin  (at  —  no)  are  called  +.  In  other  words,  the 
+  forms  lie  over  the  acute  angle  (j3,  Fig.  83)  formed 
by  the  intersection  of  the  vertical  and  the  clino  axes, 
while  the  —  forms  lie  over  the  obtuse  angle  between 
these  axes. 

It  should  be  noted  that  the  +  forms  have  a  — 
sign,  while  the  —  forms  have  a  +  sign  in  their 
symbols.  This  inconsistency  is  unfortunate,  but  the 
practice  of  naming  these  forms  in  the  manner 
specified  has  become  so  firmly  established  that  it 


118  MONOCLINIC  SYSTEM 

appears  impossible  to  change  the  nomenclature. 
The  following  rule  will  be  found  useful  in  distinguish- 
ing between  +  and  —  pyramids: 

A  +  pyramid  intersects  the  basal-pinacoid  on  top 
of  a  crystal  in  edges  which  converge  away  from  the 
observer. 

A  —  pyramid  intersects  a  basal-pinacoid  on  top 
of  a  crystal  in  edges  which  converge  toward  the 
observer. 

Reason  Why  Faces  on  Top  or  in  Front  of  a  Crystal  are 
Duplicated  at  the  Bottom  or  Back. 

It  is  easy  to  understand  why  the  clino-pinacoid 
faces  are  duplicated  on  both  sides  of  a  crystal,  since 
the  presence  of  a  symmetry  plane  requires  such 
duplication.  It  is  not  so  easy  to  understand,  how- 
ever, why  the  faces  of  the  other  forms  are  duplicated 
on  the  top  and  bottom  and  front  and  back,  since  no 
symmetry  plane  lies  between  such  duplicated  faces. 
The  reason  for  this  duplication  is  found  in  the  law  of 
axes  which  states  that  opposite  ends  of  crystal  axes 
must  be  cut  by  the  same  number  of  similar  faces 
similarly  placed.  In  order  that  this  law  shall  hold 
good  no  matter  how  the  vertical  and  clino  axes  are 
chosen,  it  is  necessary  that  faces  be  duplicated  in  the 
manner  mentioned. 

Method  of  Determining  Holohedral  Monoclinic  Forms 
by  the  Use  of  Symbols. 

The  method  outlined  in  the  presentation  of  the 
orthorhombic  system  (see  p.  107)  may  be  applied 
with  equal  facility  to  the  monoclinic  system,  al- 
though the  conception  of  +  and  —  forms  already 


HOLOHEDRAL  DIVISION  119 

outlined  must  be  borne  in  mind  when  naming  pyra- 
mids and  ortho-domes. 

Suggestions  for  Attaining  Facility  in  the  Recognition 

of  Forms. 

Orient  the  crystal  and  determine  which  of  the 
following  descriptions  apply  to  the  face  or  faces  of 
different  shape  or  size  seen. 

+  Pyramid:  A  face  whose  plane  cuts  all  three  axes 
at  finite  distances  from  the  origin,  and  the  clino  axis 
behind  the  vertical  axis. 

—  Pyramid:    A  face  whose  plane  cuts  all  three 
axes  at  finite  distances  from  the  origin,  and  the  clino 
axis  in  front  of  the  vertical  axis. 

Clino-dome:  A  face  sloping  from  the  vertical  axis 
down  to  the  right  or  left  and  parallel  to  the  clino 
axis. 

A  clino-dome  may  often  be  distinguished  with  ease 
from  the  +  or  —  pyramid  if  it  is  remembered  that 
its  faces  intersect  a  basal-pinacoid  or  another  clino- 
dome  in  edges  that  are  parallel. 

Prism:  A  vertical  face  oblique  to  the  ortho  axis. 

Ortho-pinacoid:  A  vertical  face  extending  from 
right  to  left. 

+  Ortho-dome:  A  face  whose  plane  is  parallel  to 
the  ortho  axis,  and  cuts  the  clino  axis  behind  the 
origin. 

—  Ortho-dome:    A  face  sloping  down  from  the 
vertical  axis  directly  toward  the  observer,  and  cut- 
ting the  clino  axis  in  front  of  the  origin. 

Basal-pinacoid:  A  face  sloping  down  from  the 
vertical  axis  directly  toward  the  observer  and  paral- 
lel to  the  clino  axis. 


120 


MONOCLINIC  SYSTEM 


Clino-pinacoid:  A  vertical  face  extending  from 
front  to  back. 

Fixed  and  Variable  Forms. 

The  only  holohedral  monoclinic  fixed  forms  are 
the  three  pinacoids. 

Fixed  Angles  of  the  Monoclinic  System. 

The  only  fixed  angle  in  this  system  is  that  between 
the  clino-  and  the  ortho-  or  basal-pinacoid,  namely, 
90°. 

Miscellaneous. 

The  general  statements  made  in  the  discussion  of 
the  holohedral  division  of  the  isometric  system 
regarding  combination  of  forms,  determination  of 
the  number  of  forms,  and  limiting  forms  apply  with 
equal  truth  to  this  system. 


FIG.  93.  —  Holohedral  monoclinic  crystals. 

A:  (clino  axis  selected  parallel  to  the  face  lettered  c) :  basal- 
pinacoid  (c),  —  ortho-dome  (e),  two  +  ortho-domes  (I  and  s), 
two  —  pyramids  (v  and  w),  +  pyramid  (ri),  prism  (m),  and 
ortho-pinacoid  (a). 

B:  (clino  axis  selected  parallel  to  the  face  lettered  c) :  basal- 
pinacoid  (c),  +  ortho-dome  (?/),  ortho-pinacoid  (a),  clino- 
dome  (n),  clino-pinacoid  (6),  +  pyramid  (o),  and  two  prisms 
(m  and  z). 


HOLOHEDRAL  DIVISION 


121 


Hemihedral  and  Hernimorphic  Forms. 

Hemihedral  and  hemimorphic  forms  are  too  rare 
to  require  consideration. 

Table  of  Holohedral  Monoclinic  Symbols  Used  by  Va- 
rious Authorities. 


Weiss. 

Naumann. 

Dana. 

Miller. 

+Pyramid  

—  na  :  b  :  me 

+mP  or  +mPn 

+m  or  +m-n 

hhl  or  hkl 

—Pyramid  

-f  na  :  b  :  me 

—mP  or  —  mPn 

—  m  or  —  m—n 

hhl  or  hkl 

Prism  

na  i  b  '.  oo  c 

oo  P  or  oo  Pn 

I  or  i-n 

110  or  hkO 

Clino-dome  

oo  a  :  6  :  me 

TOP  00 

m-i 

Qkl 

Ortho-pinacoid.  .. 

a  :  oo  6  :  oo  c 

°°PSo 

t-f 

100 

+Ortho-dome.  .  .  . 

—  na  '.  oo  6  :  me 

+mP«- 

+m-f 

hOl 

-Ortho-dome.  .  .  . 

-(-na  :  oo  6  :  me 

—  mP  00 

—  m-t 

Mi 

Basal-pinacoid.  .  .  . 

ooo  :  006  :  c 

OP  ^5 

o 

001 

Clino-pinacoid  

ooa  :  6  :  ooc 

ooPoo 

H 

000 

CHAPTER  VII 
TBICMNIC  SYSTEM 

HOLOHEDRAL  DIVISION 

Symmetry* 

The  triclinic  system  is  characterized  by  the  ab- 
sence of  any  kind  of  symmetry  plane. 

Selection,   Position,   and  Designation  of  the  Crystal 
Axes. 

Three  crystal  axes  are  selected  parallel  to  promi- 
nent crystallographic  directions  (see  p.  114)  and  at 
as  nearly  right  angles  to  each  other  as  possible. 
These  are  held  and  named  exactly  as  in  the  ortho- 
rhombic  system  (see  p.  103)  excepting  that  the 
macro  axis  extending  from  right  to  left  and  the  brachy 
axis  extending  from  front  to  back  will  not  be  hori~ 
zontal;  and  none  of  the  axes  are  at  right  angles  to 
each  other,  nor  can  the  angles  between  them  be  the 
fixed  angles  of  any  system.  it 

in 

Orienting  Crystals. 

The  three  crystal  axes  having  been  selected,  the 
one  chosen  as  the  c  axis  is  held  vertically;  the 
shorter  of  the  other  two  axes  (the  brachy  axis)  is 
so  held  as  to  slope  down  directly  toward  the  ob- 
server. The  macro  axis  will,  then,  extend  from 
right  to  left,  intersecting  the  plane  through  the 
vertical  and  brachy  axes  more  or  less  obliquely. 

122 


HOLOHEDRAL  DIVISION  123 

The  c  axis  is  selected  according  to  the  conven- 
tions already  given  in  the  discussion  of  the  ortho- 
rhombic  system  (see  p.  104). 

Triclinic  Forms  Discussed. 

The  triclinic  forms  have  exactly  the  same  names 
and  symbols  as  the  corresponding  orthorhombic 
forms  (see  p.  105),  but  differ  therefrom  in  that  each 
triclinic  form  consists  of  only  a  single  pair  of  paral- 
lel and  opposite  faces. 

Since  each  form  in  this  system  consists  of  but  two 
faces,  it  follows  that  the  forms  differ  from  one  an- 
other only  as  regards  their  position  with  respect  to 
the  crystal  axes;  and  any  form  may  be  changed 
into  any  other  form  by  selecting  the  crystal  axes 
so  as  to  run  in  the  proper  direction.  If,  however,- 
the  conventions  with  respect  to  the  choice  of  the 
axes  are  observed,  different  observers  will  in  most 
cases  designate  all  the  forms  by  the  same  names. 

Synonyms  for  the  Names  of  the  Triclinic  Forms. 
Pyramid  —  unit-,  brachy-,  and  macro-tetrapyra- 

mid. 

Prism  —  unit-,  brachy-,  and  macro-hemiprism. 
Macro-dome  —  hemi-macro-dome. 
Brachy-dome  —  hemi-brachy-dome. 
Macro-pinacoid  —  none. 
Brachy-pinacoid  —  none. 
Basal-pinacoid  —  basal-plane. 

Reason  Why  All  Triclinic  Forms  Consist  of  Two  Par- 
allel Faces. 

It  is  not  at  first  easy  to  see  why  a  face  on  one  side 
of  a  triclinic  crystal  must  be  duplicated  on  the 


124  TRICLINIC  SYSTEM 

opposite  side,  since  no  symmetry  plane  lies  between 
these  faces;  but  the  law  of  axes  states  that  opposite 
ends  of  crystal  axes  must  be  cut  by  the  same  number 
of  similar  faces  similarly  placed;  and,  in  order  that 
this  law  shall  hold  good  no  matter  how  the  crystal 
axes  are  chosen,  it  is  necessary  that  faces  be  dupli- 
cated in  the  manner  mentioned. 

Method  of  Determining  Triclinic  Forms  by  the  Use  of 
Symbols. 

The  method  already  outlined  in  tKe  discussion  of 
the  other  crystal  systems  may  be  applied  with  equal 
facility  to  the  triclinic  system,  in  which  it  is  not 
necessary  to  distinguish  between  +  and  —  forms. 
Care  must  be  taken  to  give  a  name  to  every  pair  of 
opposite  and  parallel  faces. 

Suggestions  for  Attaining  Facility  in  the  Recognition 
of  Forms. 

When  the  axes  are  nearly  perpendicular  to  each 
other,  it  is  possible  to  determine  the  forms  by 
slightly  modifying  the  rules  already  given  for  deter- 
mining those  of  the  same  name  in  the  orthorhombic 
system  (see  p.  108).  The  modifications  required 
are  those  introduced  by  the  fact  that  none  of  the 
axes  lie  at  right  angles  to  each  other.  In  many 
cases,  however,  the  forms  can  be  determined  most 
readily  by  noting  the  relationship  of  the  faces  with 
respect  to  the  axes,  which  is  really  equivalent  to 
determining  the  symbol  of  each  face.  The  follow- 
ing descriptions  of  the  forms  are  based  on  their  rela- 
tionship to  the  axes,  and  may  be  applied  after  a 
crystal  is  properly  oriented. 


HOLOHEDRAL  DIVISION  125 

Pyramid:  A  face  whose  plane  cuts  all  three  crys- 
tal axes  at  finite  distances  from  the  origin. 

In  order  to  determine  the  number  of  pyramids,  it 
is  necessary  to  count  all  the  pyramidal  faces  lying 
above  a  plane  passed  through  the  macro  and  brachy 
axes. 

Prism:  A  vertical  face  whose  plane  intersects 
both  the  macro  and  brachy  axes  at  finite  distances 
from  the  origin. 

In  order  to  determine  the  number  of  prisms,  it  is 
necessary  to  count  all  the  prismatic  faces  lying  in 
front  of  a  plane  passed  through  the  macro  and  verti- 
cal axes. 

Macro-dome:  A  face  whose  plane  is  parallel  to 
the  macro  axis  and  intersects  the  vertical  and  the 
brachy  axes  at  finite  distances  from  the  origin. 

In  order  to  determine  the  number  of  macro- 
domes,  it  is  necessary  to  count  all  such  faces  lying 
above  a  plane  passed  through  the  macro  and  brachy 
axes. 

Br achy-dome:  A  face  whose  plane  is  parallel  to 
the  brachy  axis  and  intersects  the  macro  and  verti- 
cal axes  at  finite  distances  from  the  origin. 

In  order  to  determine  the  number  of  brachy- 
domes,  it  is  necessary  to  count  the  number  of  such 
faces  lying  above  a  plane  passed  through  the  macro 
and  brachy  axes. 

Macro-pinacoid:  A  vertical  face  parallel  to  the 
macro  and  vertical  axes. 

There  can  be  but  one  macro-pinacoid  on  a  crystal. 

Brachy-pinacoid:  A  vertical  face  extending  from 
front  to  back  (parallel  to  the  brachy  and  vertical 
axes). 


126 


TRICLINIC  SYSTEM 


There  can  be  but  one  brachy-pinacoid  on  a 
crystal. 

Basal-pinacoid:  A  face  (not  horizontal)  parallel 
to  the  macro  and  brachy  axes. 

There  can  be  but  one  basal-pinacoid  on  a  crystal. 


FIG.  94.  —  Holohedral  triclinic  crystals. 

A:  (vertical  axis  selected  parallel  to  the  edge  between  a  and 
m,  macro  axis  parallel  to  the  edge  between  c  and  a,  and  brachy 
axis  parallel  to  the  edge  between  6  and  x) :  basal-pinacoid  (c), 
macro-pinacoid  (a),  brachy-pinacoid  (6),  brachy-dome  (x), 
pyramid  (ft),  and  prism  (m). 

B:  (vertical  axis  selected  parallel  to  the  edge  between  a  and 
M,  macro-axis  parallel  to  the  edge  between  c  and  a  at  the 
upper  right  edge  of  the  figure,  and  brachy  axis  parallel  to  the 
edge  between  c  and  6):  basal-pinacoid  (c),  macro-pinacoid 
(a),  brachy-pinacoid  (6),  four  pyramids  (p,  q,  r  and  s),  and 
five  prisms  (M,  g,  m,  n  and  o). 


In  order  to  attain  facility  in  the  recognition  of 
triclinic  forms,  it  will  be  found  advisable  to  prac- 
tice determining  them  when  the  crystal  axes  are  so 
selected  as  to  make  decidedly  acute  angles  with 
each  other.  When  the  student  is  able  to  name  the 
forms  correctly  under  such  conditions  he  will  find 
it  very  easy  to  do  so  when  the  axes  are  properly 
selected  at  as  nearly  right  angles  as  possible. 


HOLOHEDRAL  DIVISION  127 

Miscellaneous. 

The  general  statements  made  in  the  discussion  of 
the  holohedral  division  of  the  isometric  system  re- 
garding combination  of  forms,  determination  of  the 
number  of  forms,  and  limiting  forms  apply  with 
equal  truth  to  this  system. 

Hemihedral,    Tetartohedral,    and    Hemimorphic    Tri- 
clinic  Forms. 

Since  the  triclinic  system  contains  no  symmetry 
planes,  it  is  impossible  to  develop  hemihedral  or 
tetartohedral  forms  according  to  the  general  rules 
already  given  (see  p.  17).  Hemimorphic  forms  are 
practically  unknown. 


CHAPTER  VIII 

TWINS 

A  Twin  Defined. 

r\  twin  may  be  defined  as  two  or  more  crystals  or 
portions  of  one  crystal  so  united  that,  if  alternate 
crystals  or  portions  could  be  revolved  180°  on  a 
so-called  twinning  plane  or  planes,  one  simple 
untwinned  crystal  would  be  formecLJ  See  Figs.  95 
to  99. 

It  is,  of  course,  not  supposed  that  Nature  actually 
revolves  alternate  crystals  or  portions  of  a  crystal 
after  the  simple  or  untwinned  crystals  have  started 
to  form.  The  definition  just  given  is,  then,  merely 
a  statement  of  tests  which  may  be  applied  to  as- 
certain whether  a  given  crystal  or  group  of  crystals 
is  a  twin.  Since  the  cause  of  the  development  of 
twins  is  unknown,  and  since  there  are  several  classes 
differing  in  appearance,  it  is  impossible  to  formulate 
a  definition  based  either  on  genesis  or  appearance, 
but  some  authorities  define  a  twin  as  two  or  more 
crystals  or  portions  of  a  crystal  united  according  to 
some  definite  law. 

Most  twins  are  characterized  by  the  presence  of 
re-entrant  angles,  but  the  same  peculiarity  is  shown 
by  groups  of  crystals  not  united  according  to  the 
laws  of  twinning,  so  this  feature  cannot  be  considered 
distinctive  of  twins. 

128 


TWINS  129 

Preliminary  Definitions. 

QT  Twinning  Plane  Defined:  A  twinning  plane  is  a 
plane  so  located  with  reference  to  two  twinned 
crystals  or  portions  of  a  crystal  that,  if  one  of  these 
crystals  or  portions  of  a  crystal  could  be  revolved 
180°  on  the  plane,  the  two  crystals  or  portions  of  a 
crystal  would  then  be  in  untwinned  relationship  to 
each  other  (see  Fig.  95).  A  twinning  plane  is  named 
by  stating  the  naiga  of  the  possible  crystal  faces  to 
which  it  is  parallejTj 

A  twinning  plane  can  never  be  parallel  to  a  sym- 
metry plane  excepting  in  the  tetrahedral  hemihedral 
division  of  the  isometric  system  and  the  sphenoidal 
hemihedral  division  of  the  tetragonal  system,  and 
must  be  parallel  to  possible  crystal  faces. 

Twinning  Axis  Defined  :(£  twinning  axis  is  a  line 
or  direction  perpendicular  to  a  twinning  planej  The 
twinning  axis  usually  passes  through  the  geometric 
center  of  the  crystal. 

Plane  of  Union  or  Composition  Face  Defined:  The 
plane  of  union  or  composition  face  is  a  plane  along 
which  two  crystals  or  portions  of  a  crystal  appear  to 
be  united  to  form  a  twin.  It  may  or  may  not 
coincide  in  position  with  the  twinning  plane.  The 
plane  of  union  must  be  parallel  to  a  possible  crystal 
form,  and  is  named  by  stating  the  name  of  the 
possible  crystal  faces  to  which  it  is  parallel. 

Classes  of  Twins. 

Three  classes  of  twins  are  generally  recognized, 
namely,  contact,  inter  penetration,  and  multiple 
twins.  The  last  named  class  may  be  subdivided 
into  subclasses  called  oscillatory  and  cyclic  twins. 


130  TWINS 

Each  of  these  classes  will  be  discussed  in  the  order 
mentioned. 

Contact  Twin  Defined:  A  contact  twin  is  one  in 
which  two  portions  of  a  crystal  appear  to  have  been 
united  along  a  common  plane  after  one  portion  has 
been  revolved  180°  relative  to  the  other  (see  Fig.  95). 
The  twinning  plane  and  plane  of  union  usually 
coincide  in  contact  twins. 

Contact  twins  are  simpler  and  commoner  than  any 
of  the  other  types,  and  present  no  special  difficulties. 


A 

B 

FIG.  95.  —  Contact  twins.     A  is  tetragonal,  and  B  is  mono- 
clinic.     Positions  of  the  twinning  planes  indicated  by  xyz. 


In  studying  and  reciting  upon  any  type  of  twin  a 
student  should  determine  and  state  the  following 
facts : 

I.  The  class  of  the  twin  (contact,  interpenetra- 
tion,  etc.). 

II.  The  system  and  division  to  which  the  crystal 
belongs. 

III.  The  forms  present  on  the  crystal. 

IV.  The  name  of  the  form  whose  face  or  faces 
the  twinning  plane  parallels. 

V.  The  name  of  the  form  whose  face  or  faces  the 
plane  of  union  parallels. 


TWINS  131 

In  determining  the  system  of  a  contact  twin  and 
the  forms  present  thereon,  it  is  usually  advisable  to 
cover  with  the  hand  that  portion  of  the  crystal  at 
one  side  of  the  twinning  plane  and  to  examine  only 
the  portion  left  uncovered.  If  this  is  not  done,  a 
beginner  is  apt  to  be  confused  by  the  more  or  less 
unsymmetrical  arrangement  of  faces  on  the  two 
portions  of  the  crystal  separated  by  the  twinning 
plane. 

Interpenetration  Twin  Defined:  An  interpenetra- 
tion  twin  is  one  in  which  two  or  more  complete  crys- 


FIG.  96.  —  Interpenetration  twins.  A  is  orthorhombic  and 
B  is  tetrahedral  hemihedral  isometric.  Positions  of  the  twin- 
ning axes  indicated  by  X—X'. 

tals  appear  mutually  to  penetrate  into  and  through 
each  other  according  to  the  laws  of  twinning  (see 
Fig.  96). 

It  is  usually  comparatively  easy  to  determine  the 
plane  of  union  and  the  system  and  division  of  such 
twins,  together  with  the  forms  present  thereon;  but 
it  is  more  difficult  to  determine  the  name  of  the  form 
whose  face  or  faces  the  twinning  plane  parallels.  It 
will  be  found  advisable  to  seek  the  twinning  axis, 
and,  when  this  is  found,  determine  the  name  of  the 
form  with  a  face  or  faces  perpendicular  to  this  axis. 


132 


TWINS 


Such  a  form  will,  of  course,  have  faces  parallel  to  the 
twinning  plane.  To  determine  the  position  of  the 
twinning  axis,  hold  a  pencil  with  one  end  against 
various  points  on  the  crystal,  and  ascertain  whether 
it  is  possible  to  bring  all  points  on  one  of  the  inter- 
penetrating crystals  into  the  position  of  identical 
points  on  another  of  the  interpenetrating  crystals  by 
imagining  a  rotation  of  all  points  on  the  first  crystal 
180°  around  the  axis  represented  by  the  pencil.  If 
such  a  rotation  would  cause  the  two  crystals  to 
coincide,  it  may  be  assumed  that  the  pencil  is  in  the 
position  of  the  twinning  axis  sought,  provided  that 
a  plane  perpendicular  to  the  pencil  is  parallel  to  a 
possible  crystal  face. 

Multiple  Twin  Defined:  A  multiple  twin  is  one  in 
which  more  than  two  portions  of  a  single  crystal 
appear  alternatingly  to  have  been  revolved  180° 
upon  parallel  or  non-parallel  twinning  planes.  Two 
adjacent  parts  separated  by  a 
twinning  plane  possess  relation- 
ships very  similar  to  those  of 
the  two  parts  of  a  contact  twin. 

Oscillatory  Twin  Defined:   An 
oscillatory  twin  is   a  multiple 

twin  in  which  the  alternate  por-  FlG-  97-  ~  Oscillatory 

,  i  multiple  twin, 

tions  appear  to  have  been  re- 
volved   180°    upon   parallel   twinning   planes    (see 
Fig.  97). 

In  studying  multiple  twins,  only  a  portion  bounded 
on  one  side  by  a  twinning  plane  and  elsewhere  by 
crystal  faces  should  be  examined,  as  other  portions 
included  between  two  parallel  twinning  planes  are 
apt  to  possess  such  a  deficiency  of  faces  as  to  make 


TWINS 


133 


the  determination  of  the  system,  division,  and  forms 
difficult  or  impossible. 

Cyclic  Twin  Defined:  A  cyclic  twin  is  a  multiple 
twin  in  which  alternate  portions  appear  to  have  been 
revolved  180°  upon  non-parallel 
twinning  planes  (see  Fig.  98) . 

The  rotation  of  alternate  parts 
180°  on  non-parallel  twinning  planes 
tends  to  give  these  twins  a  ring-like 
form.  In  cases  where  a  small  num- 
ber of  parts  are  involved,  or  where  FlG- 
the  twinning  planes  are  nearly 


.  —  Cyclic 
multiple  twin. 


parallel,  the  ring  will  be  incomplete;  but,  when 
the  number  of  parts  are  higher  or  the  twinning 
planes  depart  considerably  from  parallelism,  a  com- 
plete ring  may  result.  The  center  of  such  a  ring  of 
twinned  portions  may  be  hollow,  or  the  twinned 
portions  may  be  in  contact  in  the  center.  If  the 
cyclic  twin  forms  an  incomplete  ring,  it  is  compara- 
tively easy  to  determine  all  the  forms  according  to 
the  method  suggested  in  the  discussion  of  oscilla- 
tory twins;  but,  if  a  complete  ring  is  formed,  some 
of  the  crystal  faces  necessary 
in  order  to  cover  completely 
an  individual  crystal  may  be 
missing. 

When   cyclic   twins   have 
the  form  of  complete  rings     FIG.  99.  —  Multiple  in- 
it  is  customary  to  give  them         terpenetration  twin, 
a  name  depending  upon  the  number  of   twinned 
portions,  as  trilling,  fourling,  sixling,  eightling,  etc. 

Interpenetration  twins  are  often  of  cyclic  type  as 
illustrated  by  Fig.  99. 


134  TWINS 

Influence  of  Twinning  Upon  Symmetry. 

The  twinning  plane  often  appears  to  be  a  sym- 
metry plane.  Whether  this  will  be  the  case  or  not 
depends  upon  the  character  of  the  crystal  and  the 
position  of  the  plane  as  set  forth  in  the  following 
statement: 

The  twinning  plane  will  always  appear  to  be  a 
symmetry  plane  if  the  crystal  is  entirely  bounded 
by  forms  whose  faces  are  arranged  in  parallel  pairs, 
an'd  if  the  twinning  plane  passes  through  the  center 
of  the  crystal. 

While  twinning  often  apparently  introduces  a 
symmetry  plane  in  a  crystal,  it  may  also  result  in 
the  elimination  of  some  symmetry  planes.  Whether 
it  will  have  the  latter  effect  or  not  depends  upon  the 
degree  of  symmetry  of  the  untwinned  crystal,  and 
the  position  of  the  twinning  plane.  It  may,  how- 
ever, be  said  that  in  general  twinned  crystals  belong- 
ing to  systems  or  divisions  of  systems  characterized 
by  the  presence  of  more  than  two  symmetry  planes 
often  appear  to  have  their  symmetry  decreased; 
while  crystals  belonging  to  systems  characterized  by 
the  presence  of  less  than  two  symmetry  planes  usu- 
ally appear  to  have  their  symmetry  increased.  For 
instance,  the  twinned  holohedral  tetragonal  crystal 
shown  in  Fig.  95A  appears  to  have  only  two  sym- 
metry planes  instead  of  five  symmetry  planes  which 
the  untwinned  crystal  possesses;  while  the  twinned 
monoclinic  crystal  shown  in  Fig.  95B  appears  to 
have  two  symmetry  planes  instead  of  the  single  one 
characteristic  of  the  untwinned  crystal. 


TWINS  135 

Possible  and  Impossible  Twinning  Planes. 

It  has  already  been  mentioned  that  a  twinning 
plane  must  be  parallel  to  a  possible  crystal  face, 
and  cannot  be  parallel  to  a  symmetry  plane.  The 
student  will  find  it  a  good  review  of  his  knowledge 
of  the  positions  of  the  symmetry  planes  and  forms 
in  each  division  of  every  system  if  he  will  attempt 
to  write  out  a  complete  list  of  all  the  forms  parallel 
to  which  twinning  planes  cannot  lie  for  the  reason 
that  such  forms  have  two  or  more  faces  parallel  to 
a  symmetry  plane  or  planes.  For  reference  pur- 
poses such  a  list  is  given  below. 

Holohedral  isometric  —  cube  and  dodecahedron. 
Tetrahedral  hemihedral  isometric  —  no  form.    (See 
statement  under  definition  of  a  twinning  plane 
on  p.  129.) 

Pentagonal  hemihedral  isometric  —  cube. 
/  Holohedral  hexagonal  —  basal-pinacoid,  1st  order 

prism,  and  2nd  order  prism. 
Rhombohedral  hemihedral  hexagonal  —  2nd  order 

prism. 

Pyramidal    hemihedral    hexagonal  —  basal-pina- 
coid. 
Trigonal  hemihedral  hexagonal  —  basal-pinacoid 

and  2nd  order  prism. 
Trapezohedral  tetartohedral  —  no  form. 
Holohedral  tetragonal  —  basal-pinacoid,  1st  order 

prism,  and  2nd  order  prism. 
Sphenoidal  hemihedral  tetragonal  —  no  form.   (See 
statement  under  definition  of  a  twinning  plane 
on  p.  129.) 

Pyramidal    hemihedral    tetragonal  —  basal-pina- 
coid. 


136  TWINS 

-  Holohedral  orthorhombic  —  macro-,  brachy-,  and 

basal-pinacoid. 
Sphenoidal  hemihedral  orthorhombic  —  no  form. 

-  Holohedral  monoclinic  —  clino-pinacoid. 
Holohedral  triclinic  —  no  form. 


CHAPTER  IX 
MISCELLANEOUS  FEATURES 

Parallelism  of  Growth. 

When  several  crystals  which  may  or  may  not  be 
in  contact  with  each  other  have  all  similar  faces 
parallel,  parallelism  of  growth  is  said  to  exist.  When 
such  parallel-growing  crystals  are  in  contact  they 
may  mutually  interpenetrate,  and  are  then  apt  to 
be  confused  with  twins  of  interpenetration  since  they 
bear  a  superficial  resemblance  to  such  twins,  and 
re-entrant  angles  are  common  on  both. 

Cases  of  parallelism  of  growth  in  which  the  crys- 
tals are  not  in  contact  are  relatively  rare,  and  are 
difficult  to  explain.  A  good  illustration  of  their 
occurrence  is  sometimes  exhibited  by  small  but  well- 
formed  chalcopyrite  crystals  dotted  over  the  sur- 
face of  crystallized  sphalerite.  More  frequently  a 
comparatively  large  crystal  appears  to  be  made  up 
of  many  smaller  ones  arranged  in  parallel  positions. 
This  is  sometimes  splendidly  shown  by  large,  rather 
rough,  octahedral  crystals  of  fluorite. 

Parallelism  of  Growth  and  Twinning  Differenti- 
ated: A  group  of  two  or  three  interpenetrating 
crystals  in  parallel  position  may,  as  has  already  been 
mentioned,  be  confused  with  a  twin  of  interpene- 
tration, but  may  be  distinguished  therefrom  by  the 
fact  that  it  is  impossible  to  find  an  axis  so  placed 
that,  if  one  crystal  could  be  revolved  180°  around 

137 


138  MISCELLANEOUS  FEATURES 

it,  the  rotated  crystal  would  exactly  coincide  in 
position  with  another  crystal.  As  it  is  not  always 
easy  to  find  the  twinning  axis,  especially  in  the  case 
of  distorted  crystals  (see  p.  144),  it  will  be  found 
easier  to  base  the  distinction  upon  the  following 
facts: 

In  cases  of  parallelism  of  growth,  all  similar  faces 
of  all  crystals  involved  will  be  in  parallel  positions. 

In  the  case  of  twins  of  interpenetration,  some 
similar  faces  of  the  crystals  involved  will  not  be  in 
parallel  positions  although  many,  perhaps  most, 
such  faces  may  be  parallel. 

Striations. 

Striations  Defined:  Striations  are  minute  terraces 
or  steps,  so  small  that  they  often  appear  lik§  lines 
etched  or  drawn  upon  natural  crystal  faces. 

Groups  of  such  lines  in  parallel  positions  are  not 
uncommon  on  natural  crystals,  and  are  often  of 
great  service  in  determining  the  degree  of  symmetry 
of  the  crystal.  They  are  due  to  three  causes, 
namely:  (1)  oscillation  of  two  or  more  crystal 
faces;  (2)  oscillatory  twinning;  and  (3)  interfer- 
ence of  two  crystals  in  contact  with  each  other. 
Each  will  be  discussed  in  the  order  named.  When- 
ever one  face  meets  another  of  different  slope  an 
edge  is  formed;  and,  if  two  such  faces  alternate 
with  each  other,  a  series  of  edges  parallel  to  the 
first  result.  Frequent  alternation  of  two  such  faces 
is  known  as  oscillation,  and  produces  many  parallel 
edges. 

Striations  Produced  by  Oscillation  of  Faces:  Na- 
ture sometimes  appears  to  be  uncertain  as  to  which 


MISCELLANEOUS  FEATURES  139 

of  two  faces  or  forms  she  prefers  to  produce,  and, 
instead  of  one  form  being  more  or  less  prominently 
modified  by  the  other,  many  small  faces  of  each 
form  alternate  with  each  other,  and  form  a  series 
of  terraces. 

As  an  illustration,  consider  a  mineral  (such  as 
pyrite)  on  crystals  of  which  the  cube  and  pentagonal 
dodecahedron  are  equally  apt  to  occur.  The  cube 
has  a  horizontal  face  on  top  of  the  crystal,  while  the 
pentagonal  dodecahedron  has  a  face  sloping  gently 
down  toward  the  observer;  and,  if  the  latter  striates 
the  former,  the  cube  face  will  be  interrupted  by  an 
indefinite  number  of  tiny  steps  or  terraces  running 
from  right  to  left,  of  which  the  horizontal  strips 
represent  the  cube,  while  the  sloping  strips  repre- 
sent the  pentagonal  dodecahedron.  If  the  width 
of  the  latter  is  very  small,  the  cubic  shape  of  the 
crystal  may  not  be  noticeably  changed,  yet  even 
then  the  resulting  striations  may  be  very  distinct, 
since  no  light  is  reflected  from  the  surfaces  belong- 
ing to  the  pentagonal  dodecahedron  when  the  cube 
surfaces  reflect  light.  In  a  similar  fashion  a  pen- 
tagonal dodecahedron  may  be  striated  by  a  cube 
if  there  is  a  strong  tendency  for  the  former  to  pre- 
dominate over  the  latter. 

Almost  any  face  is  capable  of  striating  any  other 
face  if  the  forms  involved  are  in  the  same  division 
of  a  system,  but,  in  general,  it  may  be  said  that 
those  forms  which  intersect  at  interfacial  angles  ap- 
proaching 180°  are  much  more  apt  to  striate  each 
other  than  are  those  whose  intersections  depart  con- 
siderably from  180°.  In  fact,  faces  intersecting  at 
an  angle  of  135°  rarely  striate  each  other,  and  faces 


140  MISCELLANEOUS  FEATURES 

intersecting  at  angles  of  less  than  135°  almost  never 
do  so. 

Such  striations  always  conform  strictly  to  the  sym- 
metry of  the  face  on  which  they  are  found.  That 
is,  a  group  of  striations  on  one  side  of  a  symmetry 
plane  must  be  balanced  perfectly  by  a  similar  group 
on  the  opposite  side  of  such  a  plane.  It  follows 
from  the  statements  just  made  that  striations  pro- 
duced by  oscillation  of  faces  may  cross  a  symmetry 
plane  at  right  angles  to  it,  and  may  run  parallel  to 
a  symmetry  plane  on  both  sides  of  it,  but  they  can 
never  cross  a  symmetry  plane  obliquely.  If  the  stri- 
ations on  one  portion  of  a  crystal  face  are  so  disposed 
as  to  intersect  a  symmetry  plane  obliquely,  they 
must  be  balanced  on  the  opposite  side  of  the  sym- 
metry plane  by  a  similar  group  , 

of  striations  which  meet  the      '  ^ 

first  group  at  an  angle,  and 
which  are  similarly  inclined 
to  the  symmetry  plane.  Fig. 
100,  which  shows  a  cube  face 
striated  by  a  hexoctahedron, 

illustrates  this  law. 

T,   .     , ,      .  ,  .   , .  FIG.  100.  —  Cube  face 

It  is  the  fact  that  striations  gtriated  by  ft  hexoctahe. 

must  conform  to  the  symmetry  dron.  The  positions  of 
of  the  crystals  on  which  they  the  symmetry  planes  are 
occur  that  makes  them  of  serv-  indicated  by  broken 
ice  in  determining  the  sym-  es' 
metry  of  crystals  in  cases  where  the  crystal  forms 
present  are  not  distinctive  of  any  particular  division 
of  a  system.  For  instance,  the  cube  occurs  un- 
changed in  appearance  in  all  divisions  of  the  iso- 
metric system,  but  a  cube  striated  like  Fig.  101  must 


MISCELLANEOUS  FEATURES 


141 


have  a  pentagonal  hemihedral  arrangement  of  its 
molecules,  since  it  is  evident  that  the  secondary 
symmetry  planes  are  lacking,  while  the  principal 
symmetry  planes  are  present.  Similarly,  Fig.  102 
must  represent  a  tetrahedral  hemihedral  cube,  since 


FIG.  101.  —  Cube  striated  by 
a  pentagonal  dodecahedron. 


FIG.  102.  —  Cube  striated  by 
a  positive  tetrahedron. 


the  secondary  symmetry  planes  are  present  and  the 
principal  symmetry  planes  are  absent. 

In  order  to  determine  the  name  of  any  form  stri- 
ating  another  form,  it  is  only  necessary  to  ascer- 
tain the  name  of  a  face  which  would  intersect  the 
striated  face  in  an  edge  parallel  to  one  of  the  stria- 
tions.  For  instance,  it  is  evident  that  the  pen- 
tagonal hemihedral  cube  shown  in  Fig.  101  is  stri- 
ated by  a  face  sloping  directly  toward  the  observer 
and  parallel  to  the  right  and  left  crystal  axis.  Such 
a  form  is  either  a  dodecahedron  or  a  pentagonal 
dodecahedron.  As  a  face  of  the  latter  form  will 
intersect  a  cube  face  in  a  blunter  angle  than  that 
formed  by  the  intersection  of  a  dodecahedron  and 
a  cube,  one  is  safe  in  assuming  that  in  the  case 
under  consideration  the  striations  are  produced  by 
the  oscillation  of  cube  and  pentagonal  dodecahedron 
faces. 

Striations  Produced  by  Oscillatory  Twinning:  If 
successive  twinning  planes  in  an  oscillatory  twin 


142  MISCELLANEOUS  FEATURES 

are  very  close  together,  a  series  of  ridges  and  de- 
pressions or  terraces  may  be  formed  on  some  of  the 
faces  of  a  crystal,  as  illustrated  in  Fig.  97.  When 
such  ridges  or  terraces  are  very  narrow,  they  bear 
a  very  close  resemblance  to  striations  produced  by 
oscillation  of  faces.  In  fact,  they  cannot  always  be 
distinguished  from  such  striations,  but  they  may 
differ  therefrom  in  that  twinning  striations  may 
cross  each  other  and  may  produce  a  cross-hatched 
appearance,  they  cannot  be  parallel  to  a  symmetry 
plane  (excepting  in  the  case  of  tetrahedral  hemihe- 
dral  isometric  and  sphenoidal  hemihedral  tetrago- 
nal crystals),  and  they  may  cross  a  symmetry  plane 
obliquely.  More  important  still  is  the  fact  that 
striations  produced  by  oscillation  of  faces  are  con- 
fined to  the  surface  of  a  crystal,  while  striations 
produced  by  oscillatory  twinning 
may  be  shown  equally  well  upon 
some  cleavage  faces. 

Striations  Produced  by  Inter- 
ference: When  two  crystals  de- 
velop in  contact  with  each  other 
the    surface    between    them    is 
sometimes  striated  in  an  irregu-      FIG.  103.  —  Quartz 
lar  and  peculiar  fashion,  difficult  crystal  showing  stria- 
to  describe,  but  fairly  well  illus-  tions  produced  by  in- 
trated  by  Fig.  103.     These  stri-  * 
ations  are  usually  coarse,  and  appear  to  be  utterly 
independent  of  the  symmetry  of  the  crystal. 

Cleavage. 

Cleavage  Defined:    Cleavage  is  the  result  of  a 
tendency  shown  by  many  crystalline  substances  to 


MISCELLANEOUS  FEATURES  143 

split  more  or  less  easily  parallel  to  one  or  more 
possible  crystal  faces.  In  some  cases  this  tendency 
is  so  well  developed  that  the  cleavage  surfaces  are 
almost  as  smooth  and  highly  polished  as  crystal 
faces,  with  which  cleavage  surfaces  are  sometimes 
confused. 

If  cleavage  exists  parallel  to  one  face  of  a  given 
crystal  form,  it  is  always  possible  to  develop  it  par- 
allel to  every  other  face  of  that  same  form;  and, 
not  infrequently,  cleavages  parallel  to  the  faces  of 
two  different  forms  may  be  developed  on  any  one 
crystal. 

Cleavage  surfaces  may  be  distinguished  ordi- 
narily from  crystal  faces  by  the  fact  that  they  are 
not  usually  perfectly  flat,  but  appear  to  be  covered 
with  or  made  up  from  very  thin  sheets  or  plates, 
often  with  curving  edges. 

Crystal  Habit. 

The  general  shape  assumed  by  a  crystal  is  called 
its  habit.  Among  the  commoner  terms  descriptive 
of  habits  are  the  following: 

1.  The  name  of  some  crystal  form:  For  instance, 
it  may  be  said  that  a  crystal  has  an  octahedral  habit 
when  it  has  the  general  shape  of   an  octahedron 
no  matter  how  many  other  forms  are  present,  or 
whether  the  octahedron  itself  is  actually  present  or 
absent. 

2.  Tabular  habit:  This  term  may  be  applied  to 
any  crystal  having  the  shape  of  a  tablet  —  an  ob- 
ject with  two  dimensions  much  greater  than  the 
third. 

3.  Prismatic  habit:   This  term  may  be  applied 


144  MISCELLANEOUS  FEATURES 

to  any  crystal  greatly  elongated  in  any  one  direc- 
tion no  matter  whether  that  direction  be  parallel  to 
a  prism  or  not.  An  acicular  (needle-like)  habit  is 
merely  an  extreme  development  of  a  prismatic  habit. 

Distortion  of  Crystals. 

Most  of  the  statements  already  made  relative  to 
crystals  apply  only  to  those  which  are  geometrically 
perfect,  that  is,  those  completely  bounded  on  all 
sides  with  faces  which  in  the  case  of  any  one  form  are 
identically  of  the  same  shape  and  size,  and  are 
equally  distant  from  the  center  of  the  crystal.  Any 
departure  from  this  condition  is  known  as  distortion, 
and  it  is  unfortunately  true  that  a  great  majority  of 
crystals  are  more  or  less  distorted.  Two  kinds  of 
distortion  are  recognized,  and  these  are  known, 
respectively,  as  (1)  mechanical  distortion,  and  (2) 
crystallographic  distortion. 

Mechanical  Distortion:  Mechanical  distortion  is 
produced  by  pressure  on  a  completed  crystal.  This 
may  not  only  change  the  molecular  arrangement, 
but  it  may  also  tend  to  flatten  a  crystal  and  to  bend 
or  warp  both  it  and  some  or  all  of  the  bounding  faces. 
This  alters  the  shape  and  size  of  some  or  all  of  the 
faces  and  their  distances  from  the  center  of  the 
crystal,  and  may  also  warp  the  faces  and  change  the 
angles  which  they  make  with  each  other. 

Although  mechanical  distortion  causes  a  crystal 
to  depart  from  all  the  crystallographic  laws  already 
given,  this  departure  is  often  of  so  slight  a  nature  as 
to  make  it  possible  to  make  a  fairly  accurate  guess 
as  to  the  original  appearance  of  the  crystal,  and  thus 
to  determine  the  system  to  which  it  belongs  and  the 


MISCELLANEOUS  FEATURES 


145 


forms  represented  upon  it.     Fortunately  this  type 
of  distortion  is  comparatively  uncommon. 

Crystallographic  Distortion:    Crystallographic  dis- 
tortion may  be  conceived  to  be  produced  by  moving 


FIG.  104.  —  Crystallographi- 
cally  distorted  octahedron. 


FIG.  105.  —  Crystallographi- 
cally  distorted  cubes. 


one  or  more  faces  parallel  to  themselves  to  any  extent 
either  toward  or  away  from  the  center  of  the  crystal. 
This  may  change  greatly  the  shape  and  size  of  the 
faces.  In  fact,  it  may  result  in  all  the  faces  of  any 
one  form  differing  from  each  other  in  shape  and  size. 
Fig.  104  represents  such  a  dis- 
torted octahedron,  while  Fig.  105 
represents  distorted  cubes.  In  the 
latter  case  opposite  and  parallel 
faces  are  of  the  same  shape  and 
size,  but  differ  in  these  particu-  FlG  IQG.  — Cube 
lars  from  adjacent  faces.  modified  by  a  dodec- 

Not  infrequently  a  face  seems  to  ahedron  with  one 
have  been  moved  outward  to  such  face  of  the  latter 
an  extent  as  to  have  been  com-  « should  trun- 

cate  edge  A)  sup- 
pletely  crowded  off  the   crystal.   presseci. 

This  is   illustrated   in   Fig.    106, 
and  faces  thus  destroyed  are  said  to  be  suppressed. 
Although  the  shape  and  size  of  faces  and  their 
distances  from  the  center  of  the  crystal  are  changed 


146  MISCELLANEOUS  FEATURES 

by  crystallographic  distortion,  it  is  important  to  re- 
member that  the  angles  which  such  faces  make  with 
each  other  and  with  the  crystal  axes  remain  un- 
changed, and  that  the  angles  between  edges  are  like- 
wise unaltered.  This  follows  from  the  fact  that 
there  is  no  change  in  the  arrangement  of  the 
molecules. 

From  the  statements  just  made  it  is  evident  that 
the  definitions  of  secondary  and  principal  symmetry 
planes  already  given  cannot  be  applied  to  crystal- 
lographically  distorted  crystals.  For  these,  it  is 
necessary  to  substitute  the  following  definition  of  a 
symmetry  plane: 

Any  plane  through  a  crystal  is  a  symmetry  plane 
if  there  are  approximately  the  same  number  of  faces 
on  opposite  sides  of  this  plane,  if  most  of  these  faces 
are  arranged  in  pairs  on  opposite  sides  of  and  equally 
inclined  to  this  plane,  and  if  any  two  adjacent  faces 
on  one  side  of  this  plane  are  usually  balanced  on  the 
other  side  by  two  adjacent  faces  making  identically 
the  same  angle  with  each  other.  While  this  rule  is 
not  very  rigid,  and  some  mistakes  may  be  made  in 
its  application,  these  will  be  rare  exceptions  after 
the  student  has  studied  crystals  for  some  time. 

It  is  fortunately  true  that  crystals  subject  to 
crystallographic  distortion  are  apt  to  occur  in  groups 
rather  than  in  isolated  individuals,  and  that  some 
members  of  such  groups  are  apt  to  be  much  less  dis- 
torted than  others.  In  fact,  a  little  search  will 
usually  reveal  one  or  more  crystals  almost  geometri- 
cally perfect  in  development,  and  it  is  upon  these 
that  the  attention  should  be  fixed. 

In  examining  crystallographically  distorted  crys- 


MISCELLANEOUS  FEATURES  147 

tals,  it  will  be  found  useful  to  observe  the  following 
two  rules:  (1)  All  the  faces  of  any  one  form  on  a 
crystal  will  be  of  exactly  the  same  color,  luster,  and 
smoothness;  and,  if  any  one  face  is  striated,  all  will 
be  striated  and  in  a  similar  fashion.  (2)  Where 
suppression  of  faces  has  occurred  it  is  often  impossible 
to  decide  whether  a  form  is  holohedral  or  its  hemi- 
hedral  or  tetartohedral  equivalent,  as,  for  instance, 
a  1st  order  pyramid  or  a  rhombohedron.  In  such 
cases,  always  assume  that  the  form  under  considera- 
tion is  the  one  which  would  necessitate  the  least 
suppression  of  faces. 

As  an  illustration  of  the  latter  rule,  consider  a 
hexagonal  crystal  showing  a  1st  order  prism  capped 
with  six  pyramidal  faces  in  the  1st  order  position, 
which  may  represent  either  a  1st 
order  pyramid  or  a  +  and  a  — 
rhombohedron.  .Suppose  that  a 
single  face  in  the  position  of  a 
2nd  order  pyramid  is  found  at 
one  of  the  corners  formed  by  the 


intersection    of    two    pyramidal  \r 

and  two  prismatic  faces.  Such  FlG  107.  — Quartz 
a  crystal  is  illustrated  in  Fig.  crystal  showing  sup- 
107.  The  pyramidal  face  in  the  pression  of  2nd  order 
2nd  order  position  can  evidently  trigonal  pyramid 
be  interpreted  either  as  a  2nd  faces< 
order  pyramid  or  a  2nd  order  trigonal  pyramid.  If 
we  decide  the  former  to  be  the  correct  explanation, 
we  must  assume  the  suppression  of  eleven  faces; 
while  if  we  incline  toward  the  latter  possibility,  we 
need  assume  the  suppression  of  but  five  faces. 
According  to  the  rule  just  laid  down,  we  should 


148  MISCELLANEOUS  FEATURES 

make  the  second  assumption,  and  call  the  form  a 
2nd  order  trigonal  pyramid,  making  the  crystal 
trapezohedral  tetartohedral. 

In  conclusion,  it  should  be  remembered  that  a 
majority  of  crystals  are  not  bounded  by  faces  on  all 
sides,  but  are  attached  to  some  foreign  substance  or 
to  other  crystals.  This  means  that  a  considerable 
proportion  of  the  surface  of  most  crystals  will  not 
bear  crystal  faces.  Such  crystals  can  hardly  be 
called  distorted,  but  the  condition  mentioned  natu- 
rally adds  to  the  difficulties  involved  in  their  classi- 
fication. 

Vicinal  Forms. 

The  law  of  rationality  of  parameters  already  given 
states  that  parameters  are  always  rational,  fractional 
or  whole,  small  or  infinite  numbers.  While  this  law 
holds  for  all  the  more  prominent  faces  on  crystals 
that  are  not  mechanically  distorted,  there  sometimes 
occur  on  such  faces  rather  inconspicuous  elevations, 
often  curved,  which  accord  in  form  with  the  sym- 
metry of  the  face  on  which  they  are  found,  but  which 
are  made  up  of  faces  that  have  parameters  which  are 
either  large,  irrational,  or  both  large  and  irrational. 
Forms  possessing  such  parameters  are  termed  vicinal. 
Their  cause  is  not  understood.  They  may  be 
ignored  unless  unusually  prominent,  but  their  form 
and  distribution  are  sometimes  of  service  in  deter- 
mining the  degree  of  symmetry  of  crystals. 

Etched  Figures  and  Corrosion. 

Natural  solutions  sometimes  attack  or  corrode  the 
plane  surfaces  bounding  crystals.  When  this  hap- 


MISCELLANEOUS  FEATURES  149 

pens  the  edges  may  be  rounded  and  the  faces  curved, 
and  some  or  all  faces  may  show  small  triangular, 
quadrilateral,  or  polygonal,  flat-faced  depressions  or 
pits  called  etched  figures.  These  differ  in  shape  on 
crystals  of  different  minerals  and  even  on  the  faces  of 
different  forms  on  an  individual  crystal.  In  fact,  one 
form  on  a  crystal  may  show  well-developed  etched 
figures  while  others  are  unattacked  or  merely 
smoothly  corroded.  In  all  cases,  however,  the  shape 
of  the  etched  figures  accords  with  the  symmetry  of 
the  face  on  which  they  occur,  and  a  study  of  such 
figures  will  sometimes  prove  helpful  in  determining 
the  degree  of  symmetry  of  crystals  showing  them. 


150 


MISCELLANEOUS  FEATURES 


SYSTEMS,   DIVISION   OF    SYSTEMS,    AND   FORMS 
TABULATED 


Isometric  System 


Holohedral  Forms 
Octahedron 
Trisoctahedron 
Dodecahedron 
Trapezohedron 

'  Hexahedron  (cube) 
Hexoctahedron 
Tetrahexahedron 


T.etrahedral  Hemihedral  Forms 
zfcTetrahedron 
±  Trigonal  Tristetrahedron 
itTetragonal    Tristetrahe- 

dron 
±Hextetrahedron 

Hexahedron  (cube) 

Dodecahedron 

Tetrahexahedron 

Pentagonal  Hemihedral  Forms 
Pentagonal  Dodecahedron 
Diploid 

Octahedron 


Dodecahedron 
Hexahedron  (cube) 
Trapezohedron 
Trisoctahedron 

Gyroidal  Hemihedral  Forms 

Pentagonal      Icositetrahe- 
dron 

Octahedron 

Trisoctahedron 

Dodecahedron 

Trapezohedron 

Hexahedron  (cube) 

Tetrahexahedron 


Pentagonal 
Forms 
Tetartoid 


Tetartohedral 


Octahedron 

Trisoctahedron 

Dodecahedron 

Trapezohedron 

Hexahedron  (cube) 

Tetrahexahedron 


MISCELLANEOUS  FEATURES 


151 


Hexagonal  System 


Holohedral  Forms 
1st  Order  Pyramid 
1st  Order  Prism 
2nd  Order  Pyramid 
2nd  Order  Prism 
Dihexagonal  Pyramid 
Dihexagonal  Prism 
Basal-pinacoid 

Rhombohedral    Hemihedral 

Forms 

iRhombohedron 
Hexagonal  Scalenohedron 

1st  Order  Prism 
2nd  Order  Pyramid 
2nd  Order  Prism 
Dihexagonal  Prism 
Basal-pinacoid 

Pyramidal  Hemihedral  Forms 
3rd  Order  Pyramid 
3rd  Order  Prism 

1st  Order  Pyramid 
1st  Order  Prism 
2nd  Order  Pyramid 
2nd  Order  Prism 
Basal-pinacoid 

Trigonal  Hemihedral  Forms 
±lst  Order  Trigonal  Pyr- 
amid 

±lst  Order  Trigonal  Prism 
Ditrigonal  Pyramid 
Ditrigonal  Prism 


2nd  Order  Pyramid 
2nd  Order  Prism 
Basal-pinacoid 

Trapezohedral    Hemihedral 

Forms 
Hexagonal  Trapezohedron 

1st  Order  Pyramid 
1st  Order  Prism 
2nd  Order  Pyramid 
2nd  Order  Prism 
Dihexagonal  Prism 
Basal-pinacoid 

Trapezohedral  Tetartohedral 
Forms 

iRhombohedron 

±  2nd  Order  Trigonal  Pyr- 
amid 

±2nd  Order  Trigonal 
Prism 

Trigonal  Trapezohedron 

Ditrigonal  Prism 

1st  Order  Prism 
Basal-pinacoid 

Rhombohedral      Tetartohedral 

Forms 

1st  Order  Rhombohedron 
2nd  Order  Rhombohedron 
3rd  Order  Rhombohedron 
3rd  Order  Prism 

1st  Order  Prism 
2nd  Order  Prism 
Basal-pinacoid 


152 


MISCELLANEOUS  FEATURES 


Tetragonal  System 

Holohedral  Forms 
1st  Order  Pyramid 
1st  Order  Prism 
2nd  Order  Pyramid 
2nd  Order  Prism 
Ditetragonal  Pyramid 
Ditetragonal  Prism 
Basal-pinacoid 


2nd  Order  Pyramid 
2nd  Order  Prism 
Ditetragonal  Prism 
Basal-pinacoid 

Pyramidal  Hemihedral  Forms 
3rd  Order  Pyramid 
3rd  Order  Prism 


Sphenoidal  Hemihedral  Forms 
±  Tetragonal  Sphenoid 
±Tetragonal      Scalenohe- 
dron 

1st  Order  Prism 


1st  Order  Pyramid 
1st  Order  Prism 
2nd  Order  Pyramid 
2nd  Order  Prism 
Basal-pinacoid 


Orthorhombic  System 

Holohedral  Forms  Sphenoidal  Hemihedral  Forms 
Pyramid  Orthorhombic  Sphenoid 

Prism  

Macro-dome  Prism 

Brachy-dome  Macro-dome 

Macro-pinacoid  Brachy-dome 

Brachy-pinacoid  Macro-pinacoid 

Basal-pinacoid  Brachy-pinacoid 

Basal-pinacoid 


Monoclinic  System 


Holohedral  Forms 
±Pyramid 
Prism 
Clino-dome 


Ortho-pinacoid 
rkOrtho-dome 
Basal-pinacoid 
Clino-pinacoid 


Triclinic  System 

Holohedral  Forms 
Same  as  in  the  Orthorhombic  system 


INDEX 

Page 

Acicular  Habit  Defined 144 

Amorphous  Structure  Defined 2 

Axes  (Crystal)  Defined 12 

Axes  (Crystal),  Designation  and  Use  of 13 

Axes  (Crystal),  General  Rule  for  Choosing 12 

Axes,  Distinction  Between  Crystal  and  Symmetry 12 

Beveled  Edges  Defined 10 

Cleavage  Defined 142 

Cleavage  Surfaces  and  Crystal  Faces,  Distinction  Be- 
tween   143 

Combination  of  Forms  Discussed 25 

Composition  Face  Defined 129 

Common  Symmetry   Axis    (see   Secondary  Symmetry 

Axis). 

Common  Symmetry  Plane  (see  Secondary  Symmetry 
Plane). 

Contact  Twin  Defined 130 

Corrosion  Discussed 148 

Crystal  Axes  Defined 12 

Crystal  Axes,  Designation  and  Use  of 13 

Crystal  Axes,  General  Rule  for  Choosing 12 

Crystal,  Definition  of  a 3 

Crystal  Form  and  Shape  Differentiated 16 

Crystal  Form  Defined 15 

Crystal  Habit  Defined  and  Discussed 143 

Crystalline  Structure  Defined 

Crystallography  Defined 4 

Crystallographic  Directions  (Prominent)  Defined 114 

Crystallographic  Distortion  Discussed 145 

Crystals,  Formation  of 

Crystal  System  Defined 10 

Crystal  Systems  Listed 11,  150 

Cyclic  Twin  Defined 133 

Degree  of  Symmetry  Defined 11 

153 


154  INDEX 

Page 

Distortion  (Crystallographic)  Discussed 145 

Distortion  (Mechanical)  of  Crystals  Discussed 144 

Distortion  of  Crystals  Discussed 144 

Etched  Figures  Discussed 148 

Fixed  Forms  Defined 24 

Ground-Form  Denned 13 

Habit  (Crystal)  Denned 143 

Habit  (Prismatic)  Denned 143 

Habl.  (Tabular)  Denned 143 

Hemihedral  and  Holohedral  Forms  of  Same  Shapes,  Dis- 

tL  ction  Between 35 

Hemihedral  Forms  Denned 17 

Hemimorphic  Crystals  Defined 17 

Hemimorphic  Forms,  Naming  of 70 

Holohedral  Forms  Defined 17 

Interchangeable  Symmetry  Planes  and  Axes  Defined. . .  8 

Interfacial  Angle  Defined 8 

Interpenetration  Twin  Defined 131 

Law  Governing  Combination  of  Forms 36 

Law  of  Axes 16 

Law  of  Rationality  of  Parameters 15 

Law  of  Rationality  or  Irrationality  of  Ratios  Between 

Unit  Axial  Lengths 47 

Limiting  Forms  Discussed 28 

Mechanical  Distortion  Discussed 144 

Mineral,  Definition  of 1 

Minerals,  Structure  of 1 

Molecules,  Properties  of 1 

Multiple  Twin  Defined 132 

Number  of  Forms  on  a  Crystal,  Determination  of  the . .  25 

Octant  Defined 19 

Oriented  Crystals,  Definition  of  Term 18 

Origin  Defined 12 

Oscillatory  Twin  Defined 132 

Parallelism  of  Growth  and  Twinning  Differentiated. ...  137 

Parallelism  of  Growth  Discussed 137 

Parameter  Defined 15 

Plane  of  Union  Defined 129 

Principal  Symmetry  Axis  Defined 8 

Principal  Symmetry  Plane  Defined 7 


INDEX  155 

Page 

Prismatic  Habit  Defined 143 

Prominent  Crystallographic  Directions  Denned 114 

Repetition  of  Forms  on  a  Crystal  Discussed 25 

Replaced  Edges  and  Corners  Defined 9 

Secondary  Symmetry  Axis  Defined 8 

Secondary  Symmetry  Plane  Defined 8 

Striations  Defined 138 

Striations  Produced  by  Interference  Discussed 142 

Striations  Produced  by  Oscillation  of  Faces  Discussed  .v  138 

Striations  Produced  by  Oscillatory  Twinning  Discussed  141 

Structure,  Definition  of  Amorphous &  2 

Structure,  Definition  of  Crystalline 2 

Suppressed  Faces  Defined 145 

Symbol  Defined 16 

Symbols  of  Hemihedral  Forms 32 

Symbols  of  Tetartohedral  Forms 78 

Symmetry  Axis  Defined 7 

Symmetry  Plane  Defined 5, 146 

System  (Crystal)  Defined 10 

Systems  (Crystal)  Listed 11, 150 

Tabular  Habit  Defined 143 

Tetartohedral  Forms  Defined 17 

Triangle  of  Forms  Discussed 26 

Triangle  of  Forms,  Utilization  of 27 

Truncated  Edges  and  Corners  Defined 9 

Twin  (Contact)  Defined 130 

Twin  (Cyclic)  Denned 133 

Twin  Defined 128 

Twin  (Interpenetration)  Defined 131 

Twin  (Multiple)  Defined 132 

Twin  (Oscillatory)  Defined 132 

Twinning  Axis  Defined 129 

Twinning  Plane  Defined 129 

Twinning  Planes,  Possible  and  Impossible 129, 135 

Unit-Axial  Lengths  Defined 14 

Unit-Form  Defined 13 

Variable  Forms  Defined 24 

Vicinal  Forms  Discussed 148 

Zonal  Axis  Defined 9 

Zone  Defined 9 


UNIV 


14  DAY  USE 

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